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Morrey estimates for the gradient in non-linear variational transmission problems

Luca Esposito, Lorenzo Lamberti

Abstract

We study a class of variational transmission problems driven by nonlinear energies with discontinuous coefficients across a prescribed interface. The model setting consists of integral functionals of the form \[ \mathcal{F}(u;E)=\int_Ωσ_E(x)\,F(\nabla u)\,dx, \] where the coefficient $σ_E$ takes two constant values on complementary regions separated by a $C^1$ hypersurface, and the integrand $F$ satisfies standard $p$-growth and monotonicity conditions with $p>2$. In this nonlinear variational framework, we establish local Morrey-space regularity for the gradient of local minimizers, proving that $\nabla u\in L^{2,λ}_{\mathrm{loc}}(Ω)$ for every $0\leqλ<n$, provided $2<p<\frac{2n}{n-2}$. The proof is based on quantitative decay estimates for the energy near the interface, first obtained in a flat configuration and then extended to the general case by a suitable approximation argument.

Morrey estimates for the gradient in non-linear variational transmission problems

Abstract

We study a class of variational transmission problems driven by nonlinear energies with discontinuous coefficients across a prescribed interface. The model setting consists of integral functionals of the form where the coefficient takes two constant values on complementary regions separated by a hypersurface, and the integrand satisfies standard -growth and monotonicity conditions with . In this nonlinear variational framework, we establish local Morrey-space regularity for the gradient of local minimizers, proving that for every , provided . The proof is based on quantitative decay estimates for the energy near the interface, first obtained in a flat configuration and then extended to the general case by a suitable approximation argument.
Paper Structure (7 sections, 11 theorems, 165 equations)

This paper contains 7 sections, 11 theorems, 165 equations.

Table of Contents

  1. Introduction
  2. Auxiliary results and notation
  3. Flat case
  4. Lipschitz estimate for grad u
  5. Morrey estimate in the flat case
  6. Proof of Proposition 1.3
  7. Proof of Theorem \ref{['MainThm']}

Key Result

Theorem 1.2

Let $u \in W^{1,p}(\Omega)$ be a local minimizer of the functional $\mathcal{F}(\cdot;\Omega)$ defined inMainF, and assume that $\partial E$ is a hypersurface of class $C^1$. Suppose that assumptions (H1)--(H2) are satisfied where $2<p<\frac{2n}{n-2}$. Then, for every $0 \leq \lambda < n$, Moreover, for every $\Omega' \Subset \Omega$, there exists a constant $C = C(n, p, \nu, L, \beta, E, \operat

Theorems & Definitions (23)

  • Definition 1.1
  • Theorem 1.2: Gradient regularity in Morrey spaces
  • Remark 1.3
  • Proposition 1.4
  • Remark 1.5
  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3: Standard $p$-growth bounds
  • Theorem 2.4: Higher integrability
  • Theorem 2.5: Hölder continuity
  • ...and 13 more