Abstract
We give a simple proof of the strong maximum principle for viscosity subsolutions of fully nonlinear nonhomogeneous degenerate elliptic equations on the form
under suitable assumptions allowing for non-Lipschitz growth in the gradient term. In case of smooth boundaries, we also prove a Hopf lemma, a boundary Harnack inequality, and that positive viscosity solutions vanishing on a portion of the boundary are comparable with the distance function near the boundary. Our results apply, e.g., to weak solutions of an eigenvalue problem for the variable exponent p-Laplacian.
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Adamowicz, T., Lundström, N.L.P.: The boundary Harnack inequality for variable exonent p-Lalacian, Carleson estimates, barrier functions and p(⋅)-harmonic measures. Ann. Mat. Pura Appl. 195.2, 623–658 (2016)
Aikawa, H.: Boundary Harnack principle and Martin boundary for a uniform domain. J. Math. Soc. Jpn. 53.1, 119–145 (2001)
Aikawa, H., Kilpeläinen, T., Shanmugalingam, N., Zhong, X.: Boundary Harnack principle for p-harmonic functions in smooth euclidean domains. Potential Analysis 26.3, 281–301 (2007)
Ancona, A: Principe de Harnackà la frontière et théorème de Fatou pour un opérateur elliptique dans un domaine Lipschitzien. Ann. Inst. Fourier (Grenoble) 28.4, 169–213 (1978)
Avelin, B., Julin, V.: A Carleson type inequality for fully nonlinear elliptic equations with non-Lipschitz drift term. J. Funct. Anal. 272.8, 3176–3215 (2017)
Banuelos, R., Bass, R., Burdzy, K.: Hölder domains and the boundary Harnack principle. Duke Math. J. 64.1, 195–200 (1991)
Bardi, M., Goffi, A.: New strong maximum and comparison principles for fully nonlinear degenerate elliptic PDEs, Calculus of Variations and Partial Differential Equations 58.6, Paper no.184 (2019)
Bardi, M., Da Lio, F.: On the strong maximum principle for fully nonlinear degenerate elliptic equations. Arch. Math. 73.4, 276–285 (1999)
Bardi, M., Da Lio, F.: Propagation of maxima and strong maximum principle for viscosity solutions of degenerate elliptic equations. I: Convex operators. Nonlinear Anal. Theory Methods Appl. 44.8, 991–1006 (2001)
Bardi, M., Da Lio, F.: Propagation of maxima and strong maximum principle for viscosity solutions of degenerate elliptic equations. II. Concave operators. Indiana Univ. Math. J. 52.3, 607–627 (2003)
Bass, R., Burdzy, K.: A boundary Harnack principle in twisted hölder domains. Ann. Math. 134.2(2), 253–276 (1991)
Birindelli, I., Demengel, F.: Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators. Commun. Pure Appl. Anal. 6(2), 335–366 (2007)
Birindelli, I., Galise, G., Ishii, H.: 2018, A family of degenerate elliptic operators: maximum principle and its consequences. Annales de l’Institut Henri Poincare C Analyse non lineaire 35(2), 417–441 (2018)
Caffarelli, L.A., Cabre, X.: Fully Nonlinear Elliptic Equations American Mathematical Society Colloquium Publications 43. American Mathematical Society, Providence (1995)
Caffarelli, L.A., Fabes, E., Mortola, S., Salsa, S.: Boundary behavior of nonnegative solutions of elliptic operators in divergence form. Indiana Univ. Math. J. 30.4, 621–640 (1981)
Calabi, E.: An extension of E, Hopf’s maximum principle with an application to Riemannian geometry. Duke Math. J. 25.1, 45–56 (1958)
Capuzzo-Dolcetta, I., Vitolo, A.: On the maximum principle for viscosity solutions of fully nonlinear elliptic equations in general domain. Matematiche (Catania) 62.2, 69–91 (2007)
Dahlberg, B.: On estimates of harmonic measure. Arch. Ration. Mech. Anal. 65.3, 275–288 (1977)
De Silva, D., Savin, O.: A short proof of Boundary Harnack Principle. J. Differ. Equ. 269.3, 2419–2429 (2020)
Fan, X., Zhao, Y., Zhang, Q.: A strong maximum principle for p(x)-Laplace equations. Chin. J. Contemp. Math. 24.3, 277–282 (2003)
Fleckinger-Pellé, J., Takáč, P.: Uniqueness of positive solutions for nonlinear cooperative systems with the p-Laplacian. Indiana Univ. Math. J. 43.4, 1227–1253 (1994)
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin, Heidelberg (2015)
Harjulehto, P., Hästö, P., LêÚt, V., Nuortio, M.: Overview of differential equations with non-standard growth. Nonlinear Anal. Theory Methods Appl. 72.12, 4551–4574 (2010)
Jerison, D., Kenig, C.: Boundary behavior of harmonic functions in nontangentially accessible domains. Adv. Math. 46.1, 80–147 (1982)
Julin, V.: Generalized Harnack inequality for nonhomogeneous elliptic equations. Arch. Ration. Mech. Anal. 2.216, 673–702 (2015)
Juutinen, P., Lukkari, T., Parviainen, M.: Equivalence of viscosity and weak solutions for the p(x)-Laplacian. Annales de l’Institut Henri Poincaré, Analyse Non Lineaire 27.6, 1471–1487 (2010)
Kawohl, B., Kutev, N.: Strong maximum principle for semicontinuous viscosity solutions of nonlinear partial differential equations. Arch. Math. 70.6, 470–478 (1998)
Kemper, J.T.: A boundary Harnack principle for Lipschitz domains and the principle of positive singularities. Commu. on Pure and Appl. Maths. 25, 247–255 (1972)
Lewis, J.L., Nyström, K.: Boundary behaviour for p-harmonic functions in Lipschitz and starlike Lipschitz ring domains. Annales scientifiques de l’Ecole normale supé,rieure 40.5, 765–813 (2007)
Lewis, J.L., Nyström, K.: Boundary behavior and the Martin boundary problem for p-harmonic functions in Lipschitz domains. Ann. Math. 172.3(2), 1907–1948 (2010)
Lewis, J.L., Nyström, K.: Regularity and free boundary regularity for the -Laplace operator in Reifenberg flat and Ahlfors regular domains. J. Am. Math. Soc. 25, 827–862 (2012)
Lewis, J.L., Nyström, K.: Quasi-linear PDEs and low-dimensional sets. J. Eur. Math. Soc. 20.7, 1689–1746 (2018)
Lindqvist, P: On the growth of the solutions of the differential equation ∇⋅ (|∇u|p− 2∇u) = 0 in n-dimensional space. J. Differ. Equ. 58, 307–317 (1985)
Lundström, N.L.P.: Estimates for p-harmonic functions vanishing on a flat. Nonlinear Anal. 74.18, 6852–6860 (2011)
Lundström, N.L.P.: Phragmén-lindelöf theorems and p-harmonic measures for sets near low-dimensional hyperplanes. Potential Analysis 44.2, 313–330 (2016)
Lundström, N.L.P.: Growth of subsolutions to fully nonlinear equations in halfspaces. J. Differ. Equ. 320, 143–173 (2022)
Méndez, O.: Eigenvalues and Eigenfunctions of the p(⋅)-Laplacian. A Convergence Analysis, in Jarosz, K. Contemporary Mathematics 645 Functions spaces in analytics. Trans. Am. Math. Soc., Providence (2015)
Mikayelyan, H., Shahgholian, H.: Hopf’s lemma for a class of singular/degenerate PDE’s. Ann. Acad. Sci. Fenn. Math. 40, 475–484 (2015)
Pucci, P., Serrin, J.: The Maximum Principle. Basel, Birkhäuser Basel (2007)
Sirakov, B.: Boundary Harnack estimates and quantitative strong maximum principles for uniformly elliptic PDE. Int. Math. Res. Not. 24, 7457–7482 (2018)
Wolanski, N.: Local bounds, Harnack’s inequality and hölder continuity for divergence type elliptic equations with nonstandard growth. Revista de la Unió,n Matemática Argentina 56.1, 73–105 (2015)
Wu, J.-N.: Comparisons of kernel functions, boundary Harnack principle and relative Fatou theorem on Lipschitz domains. Ann. Inst. Fourier. (Grenoble) 28.4, 147–167 (1978)
Zhang, Q.: A strong maximum principle for differential equations with nonstandard p(x)-growth conditions. J. Math. Anal. Appl. 312.1, 24–32 (2005)
Zhang, Q.: Existence and asymptotic behavior of positive solutions to p(x)-Laplacian equations with singular nonlinearities. J. Inequalities Appl. 2007, 1–9 (2007)
Acknowledgements
We thank Tomasz Adamowicz for useful discussions, ideas, and comments. Niklas Lundström and Marcus Olofsson were partially funded by the Swedish research council grant 2018-03743; this support is hereby gratefully acknowledged.
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Lundström, N.L., Olofsson, M. & Toivanen, O. Strong Maximum Principle and Boundary Estimates for Nonhomogeneous Elliptic Equations. Potential Anal 60, 425–443 (2024). https://doi.org/10.1007/s11118-022-10055-4
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DOI: https://doi.org/10.1007/s11118-022-10055-4
Keywords
- Osgood condition
- Non-Lipschitz drift
- Non-standard growth
- Variable exponent
- Fully nonlinear
- Sphere condition
- Laplace equation
- Hopf Lemma
- Boundary Harnack inequality