Local Hölder Regularity for Quasilinear Elliptic Equations with Mixed Local-Nonlocal Operators, Variable Exponents, and Weights

2025/07/02 by Apaza, Juan Pablo Alcon
#35B45 #35B65 #35D30 #35J92 #35R11 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2507.01899

Abstract

We establish local boundedness and local Hölder continuity of weak solutions to the following prototype problem: -div(|x|^{-2 β}|∇ u|^q-2 ∇ u)+(-Δ)_{p(⋅, ⋅), β}^{s(⋅, ⋅)} u=0 \text in Ω, where Ω⊂ ℝⁿ, n ≥ 2, is a bounded domain. The nonlocal operator is defined by (-Δ)_{p(⋅, ⋅), β}^{s(⋅, ⋅)} u(x):=P . V . ∫_Ω \frac|u(x)-u(y)|^{p(x, y)-2}(u(x)-u(y))|x-y|^{n+s(x, y) p(x, y)} (1)/(|x|^β|y|^β) d y Here, p: Ω× Ω→(1, ∞) and s: Ω× Ω→(0,1) are measurable functions, q:=ess_{Ω× Ω} p, and 0 ≤ β

Citations

• Global gradient regularity and a Hopf lemma for quasilinear operators of mixed local-nonlocal type
• Global gradient estimates for the mixed local and nonlocal problems with measurable nonlinearities
• On the fractional Musielak-Sobolev spaces in R^d: Embedding results & applications
• Gradient regularity in mixed local and nonlocal problems
• Gradient regularity in mixed local and nonlocal problems
• Local Hölder regularity for nonlocal equations with variable powers
• Local regularity for nonlocal equations with variable exponents
• Description of an ecological niche for a mixed local/nonlocal dispersal:\n an evolution equation and a new Neumann condition arising from the\n superposition of Brownian and L 'evy processes
• Regularity estimates for fractional orthotropic p-Laplacians of mixed order
• On the regularity theory for mixed local and nonlocal quasilinear elliptic equations
• Regularity theory for nonlocal equations with VMO coefficients
• (Non)local logistic equations with Neumann conditions
• Non-symmetric stable operators: regularity theory and integration by parts
• Semilinear elliptic equations involving mixed local and nonlocal operators
• Linear theory for a mixed operator with Neumann conditions
• Mixed local and nonlocal elliptic operators: regularity and maximum\n principles
• Hölder regularity for nonlocal double phase equations
• A-priori bounds and multiplicity of solutions for nonlinear elliptic problems involving the fractional p(⋅)-Laplacian
• Caffarelli-Kohn-Nirenberg type inequalities of fractional order with applications
• Equivalence of solutions to fractional p-Laplace type equations
• Fractional superharmonic functions and the Perron method for nonlinear\n integro-differential equations
• The obstacle problem for nonlinear integro-differential operators
• The obstacle problem for nonlinear integro-differential operators
• On the Fractional p-laplacian equations with weight and general datum
• The effect of the Hardy potential in some Calderón-Zygmund properties for the fractional Laplacian
• Harnack inequality for non-local Schrödinger operators
• Schauder estimates in generalized Hölder spaces
• Regularity theory for general stable operators
• Nonlocal Harnack inequalities
• Nonlocal Harnack inequalities
• Hölder estimates for viscosity solutions of equations of fractional p-Laplace type
• Regularity for fully nonlinear equations driven by spatial-inhomogeneous nonlocal operators
• Radial fractional Laplace operators and Hessian inequalities
• Hitchhiker's guide to the fractional Sobolev spaces
• Hitchhikerʼs guide to the fractional Sobolev spaces
• Sharp Green Function Estimates for Δ+ Δ^α/2 in C^1,1 Open Sets and Their Applications
• Regularity results for nonlocal equations by approximation
• Heat kernel estimates and Harnack inequalities for some Dirichlet forms with non-local part
• A priori Holder estimate, parabolic Harnack principle and heat kernel estimates for diffusions with jumps
• Non-local Dirichlet Forms and Symmetric Jump Processes
• Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series No. 55)
• Holder estimates for solutions of integro differential equations like the fractional laplace

Related

• Regularity results for a class of mixed local and nonlocal singular problems involving distance function
• On the regularity theory for mixed local and nonlocal quasilinear elliptic equations
• Regularity theory for mixed local-nonlocal problem involving general stable operators
• Hölder regularity of doubly nonlinear nonlocal quasilinear parabolic equations in some mixed singular-degenerate regime
• Local Hölder regularity for nonlocal equations with variable powers