Sketches of Nonuniformly Elliptic Schauder Theory

2025/09/14 by De Filippis, Cristiana
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2509.11205

Abstract

Schauder theory is a basic tool in the study of elliptic and parabolic PDEs, asserting that solutions inherit the regularity of the coefficients. It plays a central role in establishing higher regularity for solutions to a broad class of elliptic problems exhibiting ellipticity, including those involving free boundaries. In the linear setting, Schauder theory dates back to the 1920-30s and is now considered classical. Nonlinear extensions were developed in the 1980s. All these classical results are restricted to uniformly elliptic operators and heavily rely on perturbative techniques - freezing the coefficients and comparing the solution to that of a constant-coefficient problem. However, such methods fail in the nonuniformly elliptic setting, where homogeneous a priori estimates break down and standard iteration arguments no longer apply. Here we give a brief survey on recent progresses including the solution to the longstanding problem of proving the validity of Schauder estimates in the nonlinear, nonuniformly elliptic setting.

Citations

• μ-ellipticity and nonautonomous integrals
• Nonuniform ellipticity in variational problems and regularity
• Bounded minimizers of double phase problems at nearly linear growth
• Unique continuation for nonlinear variational problems
• Higher differentiability for the fractional p-Laplacian
• Scalar minimizers with maximal singular sets and lack of Meyers property
• Quantified Legendreness and the regularity of minima
• Higher Hölder regularity for the fractional p-Laplace equation in the subquadratic case
• Quasiconvex functionals of (p,q)-growth and the partial regularity of relaxed minimizers
• Singular multiple integrals and nonlinear potentials
• Singular multiple integrals and nonlinear potentials
• Lipschitz bounds for integral functionals with (p,q)-growth conditions
• Nonuniformly elliptic Schauder theory
• Hölder regularity for weak solutions to nonlocal double phase problems
• Regularity for Orlicz phase problems
• Crouzeix-Raviart finite element method for non-autonomous variational problems with Lavrentiev gap
• Quasiconvexity and partial regularity via nonlinear potentials
• H "older continuity of the minimizer of an obstacle problem with\n generalized Orlicz growth
• Growth conditions and regularity, an optimal local boundedness result
• New Examples on Lavrentiev Gap Using Fractals
• On the regularity of minima of non-autonomous functionals
• Local boundedness and Harnack inequality for solutions of linear non-uniformly elliptic equations
• Lipschitz regularity for orthotropic functionals with nonstandard growth conditions
• Higher Hölder regularity for the fractional p-Laplacian in the superquadratic case
• Regularity for general functionals with double phase
• Regularity for general functionals with double phase
• Higher Sobolev regularity for the fractional p-Laplace equation in the superquadratic case
• Universal potential estimates
• Everywhere regularity of functionals with φ-growth
• The Wiener test and potential estimates for quasilinear elliptic equations

Related

• Schauder-type estimates and applications
• The sharp growth rate in nonuniformly elliptic Schauder theory
• Nonuniformly elliptic Schauder theory
• Nonlinear potential theoretic methods in nonuniformly ellliptic problems
• Notes on Schauder estimates by scaling for elliptic PDEs in divergence form