Abstract
We prove that local weak solutions of the orthotropic p-harmonic equation are locally Lipschitz, for every \(p\ge 2\) and in every dimension. More generally, the result holds true for more degenerate equations with orthotropic structure, with right-hand sides in suitable Sobolev spaces.
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1 Introduction
1.1 The problem
In this paper, we pursue the study of the regularity of local minimizers of degenerate functionals with orthotropic structure, that we already considered in [1,2,3,4]. More precisely, for \(p\ge 2\), we consider local minimizers of the functional
and more generally of the functional
Here, \(\Omega \subset \mathbb {R}^N\) is an open set, \(N\ge 2\), and \(\delta _1, \ldots , \delta _N\) are nonnegative numbers.
A local minimizer u of the functional \(\mathfrak {F}_0\) defined in (1.1) is a local weak solution of the orthotropic p-Laplace equation
For \(p=2\), this is just the Laplace equation, which is uniformly elliptic. For \(p>2\), this looks quite similar to the usual p-Laplace equation
whose local weak solutions are local minimizers of the functional
However, as explained in [1, 2], equation (1.2) is much more degenerate. Consequently, as for the regularity of \(\nabla u\) (i.e. boundedness and continuity), the two equations are dramatically different.
In order to understand this discrepancy between the p-Laplacian and its orthotropic version, let us observe that the map \(\xi \mapsto |\xi |^p\) occuring in the definition (1.3) of \(\mathfrak {I}\) degenerates only at the origin, in the sense that its Hessian is positive definite on \(\mathbb {R}^N\setminus \{0\}\). On the contrary, the definition of the orthotropic functional \(\mathfrak {F}_0\) in (1.1) is related to the map \(\xi \mapsto \sum _{i=1}^N |\xi _i|^p\), which degenerates on an unbounded set, namely the N hyperplanes orthogonal to the coordinate axes of \(\mathbb {R}^N\).
The situation is even worse when
for the lack of ellipticity of the degenerate p-orthotropic functional arises on the larger set
As a matter of fact, the regularity theory for these very degenerate functionals is far less understood than the corresponding theory for the standard case (1.3) and its variants.
Under suitable integrability conditions on the function \(f\), we can use the classical theory for functionals with p-growth and ensure that the local minimizers of \(\mathfrak {F}_\delta \) are locally bounded and Hölder continuous, see for example [11, Theorems 7.5 & 7.6]. This theory also assures that the gradients of local minimizers lie in \(L^r_\mathrm{loc}(\Omega )\) for some \(r>p\), see [11, Theorem 6.7].
We also point out that for \(f\in L^\infty _\mathrm{loc}(\Omega )\), local minimizers of \(\mathfrak {F}_\delta \) are contained in \(W^{1,q}_\mathrm{loc}(\Omega )\), for every \(q<+\infty \) (see [3, Main Theorem]).
1.2 Main result
In this paper, we establish the optimal regularity expected for the minimizers of \(\mathfrak {F}_\delta \), namely the Lipschitz regularity.Footnote 1 More precisely, we establish the following result.
Theorem 1.1
Let \(p\ge 2\), \(f\in W^{1,h}_\mathrm{loc}(\Omega )\) for some \(h>N/2\) and let \(U\in W^{1,p}_\mathrm{loc}(\Omega )\) be a local minimizer of the functional \(\mathfrak {F}_\delta \). Then U is locally Lipschitz in \(\Omega \).
Moreover, in the case \(\delta _1=\cdots =\delta _N=0\), we have the following local scaling invariant estimate: for every ball \(B_{2R_0}\Subset \Omega \), it holds
for some \(C=C(N,p,h)>1\).
Remark 1.2
(Comparison with previous results) This result unifies and substantially extends the results on the orthotropic functional \(\mathfrak {F}_\delta \) contained in [2], where it has been established that the local minimizers of \(\mathfrak {F}_\delta \) are locally Lipschitz, provided that:
-
\(p\ge 2\), \(N=2\) and \(f\in W^{1,p'}_\mathrm{loc}(\Omega )\), see [2, Theorem A];
-
\(p\ge 4\), \(N\ge 2\) and \(f\in W^{1,\infty }_\mathrm{loc}(\Omega )\), see [2, Theorem B].
The second result was based on the so-called Bernstein’s technique, see for example [12, Proposition 2.19]. This technique had already been exploited in the pioneering paper [17] by Uralt’seva and Urdaletova, for a class of functionals which contains the orthotropic functional \(\mathfrak {F}_0\) defined in (1.1), but not its more degenerate version \(\mathfrak {F}_\delta \). Namely, the result of [17] does not cover the case when condition (1.4) is in force.
Still for the case \(\delta _1=\cdots =\delta _N=0\), an entirely different approach relying on viscosity methods has been developped in [6]. To our knowledge, both methods are limited to (at least) bounded lower order terms f.
On the contrary, [2, Theorem A] can be considered as the true ancestor to Theorem 1.1 above. Indeed, they both follow the Moser’s iteration technique, originally introduced in [16] to establish regularity for uniformly elliptic problems. However, going beyond the two-dimensional setting requires new ideas, that we will explain in Sect. 1.3 below.
In contrast to the partial results of [2, Theorems A & B], the proof of Theorem 1.1 does not depend on the dimension and does not need any additional restriction on p, apart from \(p\ge 2\). It allows unbounded lower order terms, even if the condition \(f\in W^{1,h}_\mathrm{loc}(\Omega )\) for some \(h>N/2\) is certainly not sharp. On this point, it is useful to observe that by Sobolev’s embedding we haveFootnote 2
with \(h^*\) larger than N and as close to N as desired, provided h is close to N / 2. This means that, in terms of summability, our assumption on \(f\) amounts to \(f\in L^q_\mathrm{loc}(\Omega )\) for some \(q>N\). This is exactly the sharp expected condition on f for the local minimizers to be locally Lipschitz, at least if one nurtures the (optimistic) hope that the regularity for the orthotropic p-Laplacian agrees with that for the standard p-Laplacian.Footnote 3
Our strategy to prove Theorem 1.1 relies on energy methods and integral estimates, and more precisely on ad hoc Caccioppoli-type inequalities. This only requires growth assumptions on the Lagrangian and its derivatives and can be adapted to a large class of functionals. For instance, we briefly explain in “Appendix” how to adapt our poof to the case of nonlinear lower order terms, i.e. when \(f\, u\) is replaced by a term of the form \(G(x,u)\).
Remark 1.3
We collect in this remark some interesting open issues:
-
(1)
one word about the assumption \(p\ge 2\): as explained in [1, 2], when \(\delta _1=\cdots =\delta _N=0\), the subquadratic case \(1<p<2\) is simpler in a sense. In this case, the desired Lipschitz regularity can be inferred from [8, Theorem 2.2] (see also [9, Theorem 2.7]). However, the more degenerate case (1.4) is open;
-
(2)
in [1, Main Theorem], local minimizers were proven to be \(C^1\), in the two-dimensional case, for \(1<p<\infty \) and when \(\delta _1=\cdots =\delta _N=0\). We also refer to the very recent paper [14], where a modulus of continuity for the gradient of local mimizers is exhibited. We do not know whether such a result still holds in higher dimensions;
-
(3)
in [4, Theorem 1.4], local Lipschitz regularity is established in the two-dimensional case for an orthotropic functional, with anisotropic growth conditions; that is, for the functional
$$\begin{aligned} \sum _{i=1}^2 \frac{1}{p_i}\,\int (|u_{x_i}|-\delta _i)_{+}^{p_i}\,dx +\int f\,u\,dx,\qquad \text{ with } 2\le p_1\le p_2. \end{aligned}$$For such a functional, Lipschitz regularity is open in higher dimensions, even for the case \(\delta _1=\cdots =\delta _N=0\), i.e. for the functional
$$\begin{aligned} \sum _{i=1}^2 \frac{1}{p_i}\,\int |u_{x_i}|^{p_i}\,dx +\int f\,u\,dx,\qquad \text{ with } 2\le p_1\le p_2\le \cdots \le p_N. \end{aligned}$$We point out that in this case, Lipschitz regularity in every dimension has been obtained in [17, Theorem 1] for bounded local minimizers, under the additional restrictions
$$\begin{aligned} p_1\ge 4\qquad \text{ and } \qquad p_N<2\,p_1. \end{aligned}$$Though these restrictions are not optimal, we recall that regularity can not be expected when \(p_N\) and \(p_1\) are too far apart, due to the well-known counterexamples by Giaquinta [10] and Marcellini [15].
1.3 Technical novelties of the proof
Our main result is obtained by considering a regularized problem having a unique smooth solution converging to our local minimizer, and proving a local Lipschitz estimate independent of the regularization parameter.
At first sight, the strategy to prove such an estimate may seem quite standard:
- (a):
-
differentiate equation (1.2);
- (b):
-
obtain Caccioppoli-type inequalities for convex powers of the components \(u_{x_k}\) of the gradient;
- (c):
-
derive an iterative scheme of reverse Hölder’s inequalities;
- (d):
-
iterate and obtain the desired local \(L^\infty \) estimate on \(\nabla u\).
However, steps (b) and (c) are quite involved, due to the degeneracy of our equation. This makes their concrete realization fairly intricate. Thus in order to smoothly introduce the reader to the proof, we prefer to spend some words.
We point out that our proof is not just a mere adaption of techniques used for the p-Laplace equation. Moreover, it does not even rely on the ideas developed in [2] for the two-dimensional case. In a nutshell, we need new ideas to deal with our functional in full generality.
In order to obtain “good” Caccioppoli-type inequalities for the gradient, we exploit an idea introduced in nuce in [1]. This consists in differentiating (1.2) in the direction \(x_j\) and then testing the resulting equation with a test function of the formFootnote 4
with \(1\le s\le m\). This leads to an estimate of the type (see Proposition 4.1)
Then the idea is the following: let us suppose that we are interested in improving the summability of the component \(u_{x_k}\). Ideally, we would like to take \(s=1\) in (1.6), since in this case the left-hand side boils down to
If we now sum over \(j=1,\ldots ,N\), this would give a control on the \(W^{1,2}\) norms of convex powers of \(u_{x_k}\). But there is a drawback here: indeed, this \(W^{1,2}\) norm is estimated still in terms of the Hessian of u, which is contained in the right-hand side of (1.6). Observe that (1.6) has the following form
where
This suggests to perform a finite iteration of (1.7) for \(s=s_i\) and \(m=m_i\) such that
The number \(\ell \) is chosen so that we stop the iteration when we reach \(m_\ell =0\). The above conditions imply that for every \(i=0,\ldots ,\ell \), we have
In this way, after a finite number of steps (comparable to \(\ell \)), the coupling between \(u_{x_k}\) and the Hessian of u contained in the term \(\mathcal {I}\) will disappear from the right-hand side. In other words, we will end up with an estimate of the type
Observe that we still have the Hessian of u in the right-hand side (this is the second term), but this time it is harmless. It is sufficient to use the standard Caccioppoli inequality (3.3) for the gradient, which reads
and the last term is already contained in the right-hand side of (1.8). All in all, by applying the Sobolev inequality in the left-hand side of (1.8), we get the following type of self-improving information
In this way, we obtain an iterative scheme of reverse Holder’s inequalities. This is Step 1 in the proof of Proposition 5.1 below. Thus, apparently, we safely land in step (c) of the strategy described above.
We now want to pass to step (d) and iterate infinitely many times the previous information. The goal would be to define the diverging sequence of exponents \(\gamma _\ell \) by
and conclude by iterating
Once again, there is a drawback. Indeed, observe that by definition
One may think that this is not a big issue: indeed, it would be sufficient to have
then an application of Hölder’s inequality in (1.9) would lead us to
and we could enchain all the estimates. However, since the ratio \(2^*/2\) tends to 1 as the dimension N goes to \(\infty \), it is easy to see that (1.10) cannot be true in general. More precisely, such a condition holds only up to dimension \(N=4\).
The idea is then to go back to (1.9) and use interpolation in Lebesgue spaces in order to construct a Moser’s scheme “without holes”. In a nutshell, we control the term
with
and use an iteration over shrinking radii in order to absorb the last term, see Step 2 of the proof of Proposition 5.1. Once this is done, we end up with the updated self-improving information
What we gain is that now \(2^*\,\gamma _\ell> 2\,\gamma _\ell >2\,\gamma _{\ell -1}\), thus by using Hölder’s inequality we obtain
The information comes with a precise iterative estimate and a good control on the relevant constants. We can thus launch the Moser’s iteration procedure and obtain the desired \(L^\infty \) estimate, see Step 3 of the proof of Proposition 5.1.
There is still a small detail that needs some care: the first exponent of the iteration is
which means that on \(\nabla u\) we obtain a \(L^\infty -L^{p+2}\) local estimate. Finally, in order to obtain the desired \(L^\infty -L^p\) estimate, one can simply use an interpolation argument (this is Step 4 of the proof of Proposition 5.1).
1.4 Plan of the paper
In Sect. 2, we define the approximation scheme and settle all the needed machinery. We have dedicated Sect. 3 to the new Caccioppoli inequalities which mix together the derivatives of the gradient with respect to 2 orthogonal directions. In Sect. 4, we exploit these Caccioppoli inequalities to establish integrability estimates on power functions of the gradient. In the subsequent section, we rely on these estimates to construct a Moser’s iteration scheme which finally leads to the uniform a priori estimate of Proposition 5.1.
For ease of readability, both in Sects. 4 and 5, we first consider the case \(f=0\) and \(\delta =0\), in order to emphasize the main ideas and novelties of our approach. We explain subsequently in Sects. 4.2 and 5.2 respectively the technicalities to cover the general case \(f\in W^{1,h}_\mathrm{loc}(\Omega )\) and \(\max \{\delta _i\, : \, i=1,\ldots ,N\} >0\).
Finally, in “Appendix”, we generalize Theorem 1.1 to nonlinear lower order terms.
2 Preliminaries
We will use the same approximation scheme as in [2, Section 2]. We introduce the notation
where \(0\le \delta _1,\ldots ,\delta _N\) are given real numbers and we also set
We are interested in local minimizers of the following variational integral
where \(\Omega '\Subset \Omega \) and \(f\in W^{1,h}_\mathrm{loc}(\Omega )\) for some \(h>N/2\). The latter implies that
The last inclusion is a consequence of the fact that \(p\ge 2\) and \(N\ge 2\). The condition \(f\in L^{p'}_\mathrm{loc}\) is exactly the one required in [2, Section 2] to justify the approximation scheme that we now describe.
We set
Remark 2.1
For \(p=2\) and \(\delta _i>0\), we have \(g_i\in C^{1,1}(\mathbb {R})\cap C^\infty (\mathbb {R}\setminus \{\delta _i,-\delta _i\})\), but \(g_i\) is not \(C^2\). In this case, like in [3, Section 2] one would need to replace \(g_i\) by a regularized version, in particular for the \(C^2\) regularity result of Lemma 2.2 below. In order not to overburden the presentation, we prefer to avoid to explicitely write down this regularization and keep on using the same symbol \(g_i\).
From now on, we fix U a local minimizer of \(\mathfrak {F}_\delta \). We also fix a ball
Here \(\lambda \,B\) denotes the ball having the same center as B, scaled by a factor \(\lambda >0\).
For every \(0<\varepsilon \ll 1\) and every \(x\in \overline{B}\), we set \(U_\varepsilon (x)=U*\varrho _\varepsilon (x)\), where \(\varrho _\varepsilon \) is a smooth convolution kernel, supported in a ball of radius \(\varepsilon \) centered at the origin.
Finally, we define
where \(f_{\varepsilon }=f*\varrho _{\varepsilon }\). The following preliminary result is standard, see [2, Lemma 2.5 and Lemma 2.8].
Lemma 2.2
(Basic energy estimate) There exists \(\varepsilon _0>0\) such that for every \(0<\varepsilon \le \varepsilon _0<1\), the problem
admits a unique solution \(u_\varepsilon \). Moreover, there exists a constant \(C=C(N,p)>0\) such that the following uniform estimate holds
Finally, \(u_\varepsilon \in C^{2}(B)\).
We also rely on the following stability result, which is slightly more precise than [2, Lemma 2.9].
Lemma 2.3
(Convergence to a minimizer) With the same notation as before, there exists a sequence \(\{\varepsilon _k\}_{k\in \mathbb {N}}\subset (0,\varepsilon _0)\) converging to 0, such that
where \(\widetilde{u}\) is a solution of
We also have
In the case \(\delta =1\), i.e. when \(\delta _1=\cdots =\delta _N=0\), then \(\widetilde{u}=U\) and we have the stronger convergence
Proof
The first part is proven in [2, Lemma 2.9], while (2.4) is proven in [2, Lemma 2.3]. For the case \(\delta =1\), we observe that \(\widetilde{u}=U\) follows from the strict convexity of the functional, together with the local minimality of U. In order to prove (2.5), we observe that
We now use that \(\{u_{\varepsilon _k}\}_{k\in \mathbb {N}}\) strongly converges in \(L^p(B)\), is bounded in \(W^{1,p}(B)\) and that \(\{f_{\varepsilon _k}\}_{k\in \mathbb {N}}\) strongly converges in \(L^{p'}(B)\) to f. By further using that (see the proof of [2, Lemma 2.9])
we finally get
Observe that by Clarkson’s inequality for \(p\ge 2\), we obtain
By using this and (2.6), we eventually get (2.5). \(\square \)
Remark 2.4
Observe that the functional \(\mathfrak {F}_\delta \) is not strictly convex when \(\delta >1\). Thus property (2.4) is useful in order to transfer a Lipschitz estimate for the minimizer \(\widetilde{u}\) selected in the limit, to the chosen one U.
Finally, we will repeatedly use the following classical result, see [11, Lemma 6.1] for a proof.
Lemma 2.5
Let \(0<r<R\) and let \(Z(t):[r,R]\rightarrow [0,\infty )\) be a bounded function. Assume that for \(r\le t<s\le R\) we have
with \(\mathcal {A},\mathcal {B},\mathcal {C}\ge 0\), \(\alpha _0\ge \beta _0>0\) and \(0\le \vartheta <1\). Then we have
where \(\lambda \) is any number such that
3 Caccioppoli-type inequalities
The solution \(u_\varepsilon \) of the regularized problem (2.3) satisfies the Euler-Lagrange equation
From now on, in order to simplify the notation, we will systematically forget the subscript \(\varepsilon \) on \(u_\varepsilon \) and \(f_\varepsilon \) and simply write u and \(f\) respectively.
We now insert a test function of the form \(\varphi =\psi _{x_j}\in W^{1,p}_0(B)\) in (3.1), compactly supported in B. Then an integration by parts yields
for \(j=1,\ldots ,N\). This is the equation solved by \(u_{x_j}\).
We refer to [2, Lemma 3.2] for a proof of the following Caccioppoli inequality:
Lemma 3.1
Let \(\Phi :\mathbb {R}\rightarrow \mathbb {R}^+\) be a \(C^1\) convex function. Then there exists a universal constant \(C>0\) such that for every function \(\eta \in C^\infty _0(B)\) and every \(j=1,\ldots ,N\), we have
We need a more elaborate Caccioppoli-type inequality for the gradient, which is reminiscent of [1, Proposition 3.1].
Proposition 3.2
(Weird Caccioppoli inequality) Let \(\Phi , \Psi :[0,+\infty )\rightarrow [0,+\infty )\) be two non-decreasing continuous functions. We further assume that \(\Psi \) is convex and \(C^1\). Then there exists a universal constant \(C>0\) such that for every \(\eta \in C^\infty _0(B)\), \(0\le \theta \le 2\) and \(k,j=1,\dots ,N\),
where
Proof
By a standard approximation argument, one can assume that \(\Phi \) is \(C^1\) as well. We take in (3.2)
This gives
We now proceed to estimating the three terms \(\mathcal {A}_\ell \). We have
and the integral containing the Hessian of u can be absorbed in the left-hand side of (3.5). Using also that \(2\,u_{x_j}^2\,\Phi '(u_{x_j}^2) \ge 0\), this yields
We now estimate \(\mathcal {A}_2\), which is the most delicate term: writing \(\Psi '(u_{x_k}^2)\!=\!\Psi '(u_{x_k}^2)^{\frac{\theta }{2}}\,\Psi '(u_{x_k}^2)^{1-\frac{\theta }{2}}\) and using the Cauchy-Schwarz inequality, we get
We observe that
where \(G\) is the convex nonnegative \(C^1\) function defined by
Thus by the Caccioppoli inequality (3.3) with \(x_k\) in place of \(x_j\) and
we get
By Jensen’s inequality
Together with the fact that \(G'(u_{x_k})=2\,u_{x_k}\Psi '(u_{x_k}^2)^{1-\frac{\theta }{2}}\), this implies
which in turn yields by (3.6) and (3.7),
Here, we have also used the inequality \((A+B)^{1/2} \le A^{1/2} + B^{1/2}.\)
Finally,
This completes the proof. \(\square \)
4 Local energy estimates for the regularized problem
In order to emphasize the main ideas of the proof, we have divided this section in two parts. In the first one, we explain how (3.4) leads to higher integrability estimates for the gradient when \(f=0\) and \(\delta =1\). This allows to ignore a certain amount of technicalities. In the second part, we then detail the modifications of the proof to obtain the corresponding estimates in the general case.
4.1 The homogeneous case
In this subsection, we assume that \(f=0\) and \(\delta =1\). Then the two terms \(\mathcal {E}_1(f)\) and \(\mathcal {E}_2(f)\) in (3.4) vanish. Also observe that in this case from (2.2) we have
Let us single out a particular case of Proposition 3.2 by taking
with \(1\le s \le m\).
Proposition 4.1
(Staircase to the full Caccioppoli) Let \(p\ge 2\) and let \(\eta \in C^{\infty }_0(B)\), then for every \(k,j=1, \ldots , N\) and \(1\le s\le m\)
Proof
We use (3.4) with the choices (4.1) above and
This gives
We use Young’s inequality in the form \(C\,\sqrt{a\,b}\le C^2\, b/4 +a\) for the product in the right-hand side to get
In the first term of the right-hand side, we use Young’s inequality with the exponents
We also observe for the second term that \(m^\theta \le m\). This gives the desired estimate. \(\square \)
Proposition 4.2
(Caccioppoli for power functions of the gradient) We fix an exponent
Let \(\eta \in C^{\infty }_0(B)\), then for every \(k=1, \ldots , N\) we have
for some \(C=C(N,p)>0\).
Proof
We define the two finite families of indices \(\{s_\ell \}\) and \(\{m_\ell \}\) such that
Observe that
and
In terms of these families, inequality (4.2) implies for every \(\ell \in \{0,\ldots ,\ell _0-1\}\)
for some \(C>0\) universal. By starting from \(\ell =0\) and iterating the previous estimate up to \(\ell =\ell _0-1\), we then get
for a universal constant \(C>0\). For the last term, we apply the Caccioppoli inequality (3.3) with
thus we get
that is,
possibly for a different universal constant \(C>0\).
We now observe that \(g_{i,\varepsilon }''(u_{x_i}) =\Big ((p-1)\,|u_{x_i}|^{p-2}+\varepsilon \Big )\) and thus
When we sum over \(j=1,\ldots ,N\), we get
This proves the desired inequality. \(\square \)
4.2 The non-homogeneous case
In the general case where \(f\not =0\) and/or \(\delta >1\), we can prove the following analogue of (4.2), in a similar way:
We then deduce the following analogue of Proposition 4.2:
Proposition 4.3
We fix an exponent
Let \(\eta \in C^{\infty }_0(\Omega )\), then for every \(k=1, \ldots , N\) we have
for some \(C=C(N,p)>0\).
Proof
Using the same notation and the same strategy as in the proof of (4.3), except that we start from (4.5) instead of (4.2), we get the following analogue of (4.4):
We now observe that
Noting that
we have
We deduce therefrom
thus when we sum over \(j=1,\ldots ,N,\) we get
This proves the desired inequality (4.6). \(\square \)
5 Proof of Theorem 1.1
Proof
The core of the proof of Theorem 1.1 is the uniform Lipschitz estimate of Proposition 5.1 below. Its proof, which is postponed for ease of readability, uses the integrability estimates of Sect. 4. Once we have this uniform estimate, we can reproduce the proof of [2, Theorem A] and prove that \(\nabla U\in L^\infty (\Omega ')\), for every \(\Omega '\Subset \Omega \).
We now detail how to obtain the scaling invariant local estimate (1.5) in the case \(\delta _1=\cdots =\delta _N=0\). We take \(0<r_0<R_0\le 1\) and a ball \(B_{2R_0}\Subset \Omega \). We then consider the sequence of miminizers \(\{u_{\varepsilon _k}\}_{k\in \mathbb {N}}\) of (2.3) obtained in Lemma 2.3, with B a ball slightly larger than \(B_{R_0}\) so that \(2\,B\Subset \Omega \). By using the uniform Lipschitz estimate (5.3) below, taking the limit as k goes to \(\infty \) and using the strong convergence of Lemma 2.3, we obtain
Without loss of generality, we can assume that \(\Vert \nabla U\Vert _{L^{p}(B_{R_0})}>0\). Hence, by Young’s inequality,
possibly for a different \(C=C(N,p,h)>0\). We now observe that for every \(\lambda >0\), \(\lambda \,U\) is still a solution of the orthotropic \(p-\)Laplace equation, with the right hand side \(f\) replaced by \(\lambda ^{p-1}\,f\). We can use (5.1) for \(\lambda \, U\) and get
Dividing by \(\lambda \), we obtain
We take
and observe that if \(\Vert \nabla f\Vert _{L^{h}(B_{R_0})}>0\), then
while the inequality is obvious when \(\Vert \nabla f\Vert _{L^{h}(B_{R_0})}=0\). Similarly,
It thus follows that
We now make this estimate dimensionally correct. Given \(R_0>0\), we consider a ball \(B_{2R_0}\Subset \Omega \). Then the rescaled function
is a solution of the orthotropic p-Laplace equation, with right-hand side \(f_{R_0}(x):=R_{0}^p\,f(R_0\,x)\). We can use for it the estimate (5.2) with radii 1 and 1 / 2. By scaling back, we thus obtain
for some constant \(C=C(N,p,h)>1\). Dividing by \(R_0\), we get
This concludes the proof. \(\square \)
Proposition 5.1
(Uniform Lipschitz estimate) Let \(p\ge 2\), \(h>N/2\) and \(0<\varepsilon \le \varepsilon _0\). For every \(B_{r_0}\subset B_{R_0}\Subset B\) with \(0<r_0<R_0\le 1\), we have
where \(C=C(N,p,h, \delta )>1\) and \(\sigma _i=\sigma _i(N,p,h)>0\), for \(i=1,2\).
5.1 Proof of Proposition 5.1: the homogeneous case
In this subsection, we assume that \(f=0\) and \(\delta =1\).
For simplicity, we assume throughout the proof that \(N\ge 3\), so in this case the Sobolev exponent \(2^*\) is finite. The case \(N=2\) can be treated with minor modifications and is left to the reader. For ease of readability, we divide the proof into four steps.
Step 1: a first iterative scheme We add on both sides of inequality (4.3) the term
We thus obtain
An application of the Sobolev inequality leads to
We now sum over \(k=1,\ldots ,N\) and use that by the Minkowski inequality,
This implies
We now introduce the function
We use that
and also that \(g_{i,\varepsilon }''(u_{x_i})\, |u_{x_k}|^{2\,q+2}\le C\,\mathcal {U}^{2\,q+p}+\varepsilon \,\mathcal {U}^{2\,q+2}\) for every \(1\le i, k \le N\). This yields
for a possibly different \(C=C(N,p)>1\). By using that \(\mathcal {U}^{2\,q+2}\le 1 +\mathcal {U}^{2\,q+p}\), we obtain (for \(\varepsilon <1\))
We fix two concentric balls \(B_r\subset B_R \Subset B\) and \(0<r<R\le 1\). Let us assume for simplicity that all the balls are centered at the origin. Then for every pair of radius \(r\le t<s\le R\) we take in (5.5) a standard cut-off function
This yields
We define the sequence of exponents
and take in (5.7) \(q=2^{j+1}-1\). This gives
for a possibly different constant \(C=C(N,p)>1\).
Step 2: filling the gaps We now observe that
By interpolation in Lebesgue spaces, we obtain
where \(0<\tau _j<1\) is given by
We now rely on (5.8) to get
The sequence \((\tau _j)_{j\ge 1}\) is decreasing, which implies
Hence,
Using that \(s\le R\le 1\) and \(C>1\), this implies that
By Young’s inequality,
By applying Lemma 2.5 with
we finally obtain
for some \(C=C(N,p)>1\).
Step 3: Moser’s iteration We now want to iterate the previous estimate on a sequence of shrinking balls. We fix two radii \(0<r<R\le 1\), then we consider the sequence
and we apply (5.9) with \(R_{j+1}<R_j\) instead of \(r<R\). Thus we get
where the constant \(C>1\) only depends on N and p.
We introduce the notation
thus (5.10) rewrites as
Here, we have used again that \(R\le 1\), so that the term multiplying \(Y_j\) is larger than 1. By iterating the previous estimate starting from \(j=1\) and using some standard manipulations, we obtain
possibly for a different constant \(C=C(N,p)>1\). We now take the power \(1/\gamma _n\) on both sides:
We observe that \(\gamma _{j}\sim 2^{j+2} \) as \(j\) goes to \(\infty \). This implies the convergence of the series above and we thus get
for some \(C=C(N,p)>{ 1}\) and \(\beta '=\beta '(N,p)>0\). We also used that \(\gamma _0=p+2\). By recalling the definition of \(\mathcal {U}\), we finally obtain
Step 4 \(L^\infty -L^p\) estimate We fix two concentric balls \(B_{r_0}\subset B_{R_0}\Subset B\) with \(R_0\le 1\). Then for every \(r_0\le t<s\le R_0\) from (5.11) we have
where we also used the subadditivity of \(\tau \mapsto \tau ^{1/(p+2)}\). We now observe that
We can apply again Lemma 2.5, this time with the choices
This yields
for every \(R_0\le 1\). This readily implies the desired estimate (5.3) in the homogeneous case. \(\square \)
5.2 Proof of Proposition 5.1: the non-homogeneous case
We follow step by step the proof of the homogeneous case and we only indicate the main changes, which essentially occur in Step 1 and Step 2.
Step 1: a first iterative scheme This time, we add on both sides of inequality (4.6) the term
Then the left-hand side is greater, up to a constant, than
The latter in turn, by the Sobolev inequality is greater, up to a constant, than
By summing over \(k=1,\ldots ,N\) and using the Minkowski inequality, we obtain the analogue of (5.4), namely
We now introduce the function
where the parameter \(\delta \) is defined in (2.1). We use that
and also that for every \(1\le i \le N\),
This yields
for a possibly different \(C=C(N,p, \delta )>1\).
With the concentric balls \(B_r\subset B_t \subset B_s \subset B_R\) and the function \(\eta \) as defined in (5.6), an application of Hölder’s inequality leads to
From now on, we assume that
This in particular implies that
then by using Hölder’s inequality and taking into account that \(s\le 1\), we get
Thanks to the relation on the exponents, this gives (recall that \(\varepsilon <1\) and \(s\le 1\))
We now estimate
Observe that on the set \(\{\mathcal {U}\ge 1\}\), we have \(\mathcal {U}\le 2\,\left( \mathcal {U}-1/2\right) _+\). Hence,
where the exponent \((2\,q+2)\,h'-(2^*p)/2\) is positive, thanks to the choice (5.13) of q. We deduce from (5.14) that
for a constant \(C=C(N,p,h,\delta )>1\). We now take \(q=2^{j+1}-1\) for \(j\ge j_0-1\), where \(j_0\in \mathbb {N}\) is chosen so as to ensure condition (5.13). Then we define the sequence of positive exponents
and
In order to simplify the notation, we also introduce the absolutely continuous measure
From (5.16), we get
We now observe that \(h>N/2\) implies \(h'<2^*/2\). By recalling that \(p\ge 2\), we thus have \(2\,h'<(2^*\,p)/2\), which in turn implies
It follows that
Hence, we obtain
Step 2: filling the gaps Since
we obtain by interpolation in Lebesgue spaces,
where \(0<\tau _j<1\) is given by
We now rely on (5.18) to get
We claim that
We already know by (5.17) that \((\widehat{\gamma }_j/\gamma _j) \ge 2^*/(2h')\). Moreover, relying on the fact that \((2^*\,p)/2\le 2^{j_0}\,h'\) (this follows from the definition of \(j_0\)), we also have
By recalling the definition (5.19) of \(\tau _j\), we get
Observe that on \([2^*/(2\,h'),+\infty )\times [2,4]\), the function \(x\mapsto \zeta (x,y)\) is increasing, while \(y\mapsto \zeta (x,y)\) is decreasing. Thus we get
which is exactly claim (5.21). We deduce from (5.21) and (5.17) that
In particular, we have
since the quantity inside the parenthesis is larger than \(1\) (here, we use again that \(s\le 1\)). In view of (5.20), this implies
By Young’s inequality,
where \(C=C(N,p,h,\delta )>1\) as usual. By applying again Lemma 2.5, this times with the choices
we finally obtain
Step 3: Moser’s iteration Estimate (5.22) is the analogue of (5.9), except that the Lebesgue measure \(dx\) is now replaced by the measure \(d\mu \), and the index \(j\) is assumed to be larger than some \(j_0+1\), instead of \(j\ge 0\) as in (5.9). Following the same iteration argument and starting from \(j=j_0+1\), we are led to
for some \(C=C(N,p,h, \delta )>1\), \(\beta '=\beta '(N,p,h)>0\).
Step 4 \(L^{\infty }-L^{p}\) estimate We now want to replace the norm \(L^{\gamma _{j_0}}(B_R,d\mu )\) of \(\mathcal {U}\) in the right-hand side of (5.23) by its norm \(L^p(B_R,dx)\). Let \(q_1:=2^{j_1+1}-1\) where
Then \(\gamma _{j_0}\le 2^*\,q_1\) and thus, by using that
we have
We rely on (5.14) with \(q=q_1\) to get for every \(0<r<t<s<R\)
for some new constant \(C=C(N,p,h,\delta )>1\).
Since \(j_1\ge j_0\), we have \(p<(2\,q_1+2)\,h'<(2q_1+p)\frac{2^*}{2}\), and thus, by interpolation in Lebesgue spaces
where \(\theta \in (0,1)\) is determined as usual by scale invariance. As in the proof of (5.15), we have
Inserting this last estimate into (5.26), we obtain
up to changing the constant \(C=C(N,p,h,\delta )>1\). In view of (5.25), this gives
By Young’s inequality, we get
By Lemma 2.5, this implies
after some standard manipulations. Coming back to (5.23) and taking into account (5.24), we obtain
where \(C=C(N,p,h,\delta )>1\) and \(\sigma _i=\sigma _i(N,p,h)>0\), for \(i=1,2\). By definition of \(\mathcal {U}\), we have
Since \(\Vert \mathcal {U}\Vert _{L^{\infty }(B_{r_0},\,d\mu )}+1\ge \Vert \mathcal {U}\Vert _{L^{\infty }(B_{r_0})}\), it follows that
possibly for a different constant \(C=C(N,p,h,\delta )>1\). This completes the proof. \(\square \)
Notes
Observe that when \(f\equiv 0\), any Lipschitz function u with \(|\nabla u|\le \min \{\delta _i\, :\, i=1,\ldots ,N\}\) is a local minimizer of \(\mathfrak {F}_\delta \). Thus in general Lipschitz continuity is the best regularity one can hope for.
We recall that
$$\begin{aligned} h^*=\left\{ \begin{array}{ll} N\,h/(N-h),&{} \text{ if } \,h<N,\\ \text{ any } q<+\infty , &{} \text{ if } \,h=N,\\ +\infty ,&{} \text{ if } \,h>N. \end{array}\right. \end{aligned}$$In the case of the standard p-Laplacian, the sharp assumption to have Lipschitz regularity is that f belongs to the Lorentz space \(L^{N,1}_\mathrm{loc}\). This sharp condition has been first detected by Duzaar and Mingione in [7, Theorem 1.2], see also [13, Corollary 1.6] for a more general and refined result. This sharp result is obtained by using potential estimates techniques. We recall that \(L^q_\mathrm{loc}\subset L^{N,1}_\mathrm{loc}\) for every \(q>N\) and under this slightly stronger assumption on f, Lipschitz regularity for the p-Laplacian can be proved by more standard techniques based on Moser’s iteration, see for example [5].
This test function is not really admissible, since it is not compactly supported. Actually, to make it admissible, we have to multiply it by a cut-off function. However, this gives unessential modifications and we prefer to avoid it in order to neatly present the idea of the proof.
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Acknowledgements
The paper has been partially written during a visit of P. B. & L. B. to Napoli and of C. L. to Ferrara. Both visits have been funded by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) through the project “Regolarità per operatori degeneri con crescite generali ”. A further visit of P. B. to Ferrara in April 2017 has been the occasion to finalize the work. Hosting institutions are gratefully acknowledged. The last three authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
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Communicated by L. Ambrosio.
Appendix: Lipschitz regularity with a nonlinear lower order term
Appendix: Lipschitz regularity with a nonlinear lower order term
In this section, we consider the functional
The lower order term \(f\,u\) of the functional \(\mathfrak {F}_\delta \) is thus replaced by a more general term \(G(x,u)\). We assume that \(G\) is a Carathéodory function and that for almost every \(x\in \Omega \), the map
We denote \(f(x,\xi ):=G_\xi (x,\xi )\) and we assume that \(f\in W^{1,h}_\mathrm{loc}(\Omega \times \mathbb {R})\), for some \(h>N/2\). Finally, we assume that \(G(x,\xi )\) satisfies the inequality
where \(1<p\le \gamma <p^*\) and \(a, b\) are two non-negative functions belonging respectively to \(L^{s}_\mathrm{loc}(\Omega )\) and \(L^{\sigma }_\mathrm{loc}(\Omega )\) with \(s>N/p\) and \(\sigma >p^*/(p^*-\gamma )\).
Under assumption (A.1), all the local minimizers of \(\mathfrak {G}_\delta \) are locally bounded, see [11, Theorem 7.5] and moreover, for every such minimizer \(u\), for every \(B_{r_0}\Subset B_{R_0} \Subset \Omega \),
where \(M\) depends on \(\Vert u\Vert _{W^{1,p}(B_{R_0})}, r_0, R_0, \Vert b\Vert _{L^{\sigma }(R_0)}\), and \(\Vert a\Vert _{L^{s}(B_{R_0})}\).
Then we have:
Theorem A.1
Let \(p\ge 2\) and let \({U}\in W^{1,p}_\mathrm{loc}(\Omega )\) be a local minimizer of the functional \(\mathfrak {G}_\delta \). Then U is locally Lipschitz in \(\Omega \).
Proof
We only explain the main differences with respect to the proof of Theorem 1.1. Since \(G\) is convex with respect to the second variable, the functional \(\mathfrak {G}\) is still convex. This implies that Lemma 2.3 remains true with the same proof. We then introduce the approximation of \(G\):
where \(\rho _{\varepsilon }\) is the same regularization kernel as before, while \(\widetilde{\rho }_{\varepsilon }\) is a regularization kernel on \(\mathbb {R}\).
Given a local minimizer \(U\in W^{1,p}_\mathrm{loc}(\Omega )\) and a ball \(B\subset 2\,B\Subset \Omega \), there exists a unique \(C^{2}\) solution \(u_{\varepsilon }\) to the regularized problem
where
and \(U_\varepsilon =U*\rho _{\varepsilon }\). Moreover, by [11, Remark 7.6] we have \(u_\varepsilon \in L^\infty (B)\), with a bound on the \(L^\infty \) norm uniform in \(\varepsilon >0\). In order to simplify the notation, we simply write as usual \(u\) and \(f\) instead of \(u_{\varepsilon }\) and \(f_\varepsilon \). The Euler equation is now
When we differentiate the Euler equation with respect to some direction \(x_j\), we obtain
We can then repeat the proof of Proposition 5.1 with this additional term \(f_{\xi }(x,u)u_{x_j}\) which leads to the following analogue of (5.12):
Using again Hölder’s inequality for the first three terms, we obtain inequality (5.14) where \(\Vert \nabla f\Vert \) now represents the full gradient of \(f\) with respect to both \(x\) and \(\xi \). The rest of the proof is the same and leads to a uniform Lipschitz estimate, as desired. \(\square \)
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Bousquet, P., Brasco, L., Leone, C. et al. On the Lipschitz character of orthotropic p-harmonic functions. Calc. Var. 57, 88 (2018). https://doi.org/10.1007/s00526-018-1349-3
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DOI: https://doi.org/10.1007/s00526-018-1349-3