Global Lipschitz regularity in anisotropic elliptic problems with natural gradient growth

2025/07/19 by Carlo Alberto Antonini, Antonini, Carlo Alberto, Andrea Cianchi +1
Mathematics · #35J25 #35J60 #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2507.14606

openalex publication_date 2025/07/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We deal with homogeneous Dirichlet and Neumann boundary-value problems for anisotropic elliptic operators of p-Laplace type. They emerge as Euler-Lagrange equations of integral functionals of the Calculus of Variations built upon possibly anisotropic norms of the gradient of trial functions. We establish global Lipschitz regularity of solutions under the weakest possible assumption on right-hand side of the equation, which may also include the gradient term with natural growth exponent. The results hold in either convex domains, or domains enjoying minimal integrability assumptions on the curvature of its boundary.

Citations

• Second-order boundary estimates for solutions to a class of quasilinear elliptic equations
• Weak solutions of Anisotropic (and crystalline) inverse mean curvature flow as limits of p-capacitary potentials
• Smooth approximation of Lipschitz domains, weak curvatures and isocapacitary estimates
• Asymptotic behaviour of solutions to the anisotropic doubly critical equation
• Global second-order estimates in anisotropic elliptic problems
• Liouville-type results for some quasilinear anisotropic elliptic equations
• Lipschitz regularity for solutions of a general class of elliptic equations
• Borderline global regularity for nonuniformly elliptic systems
• Interior regularity results for inhomogeneous anisotropic quasilinear equations
• Gradient estimates for singular p-Laplace type equations with measure data
• Symmetry results for critical anisotropic p-Laplacian equations in\n convex cones
• Fully anisotropic elliptic problems with minimally integrable data
• Motion of level sets by inverse anisotropic mean curvature
• Elliptic equations involving the p-Laplacian and a gradient term\n having natural growth
• A nonlinear Stein theorem
• Global boundedness of the gradient for a class of nonlinear elliptic\n systems
• Boundary regularity for solutions of degenerate elliptic equations

Related

• Global second-order estimates in anisotropic elliptic problems
• Lipschitz regularity for anisotropic fully nonlinear equations with nonstandard growth
• Lipschitz regularity results for nonlinear strictly elliptic equations and applications
• Lipschitz regularity for solutions of a general class of elliptic equations
• Lipschitz bounds for integral functionals with (p,q)-growth conditions