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Double phase meets Muckenhoupt

Daviti Adamadze, Lars Diening, Tengiz Kopaliani, Jihoon Ok

TL;DR

The paper tackles Hölder regularity for double-phase variational problems with minimal regularity on the modulating coefficient by developing a Muckenhoupt-type condition $\phi\in\mathcal{A}$ in generalized Orlicz spaces. It builds a complete analytic framework, including a generalized Jensen inequality, maximal operator boundedness, density results, and Sobolev–Poincaré-type inequalities, enabling a De Giorgi–type regularity theory. Under $\phi\in\mathcal{A}$, weak solutions $u\in W^{1,\phi}(\Omega)$ are locally Hölder continuous, with explicit oscillation decay and Hölder exponent depending on $n$, $p$, $q$, and the $\mathcal{A}$-constant. The work unifies and extends classical results (in the spirit of FKS) to non-homogeneous double-phase models, even with discontinuous modulating coefficients, advancing the regularity theory for variational problems with nonstandard growth. It also provides density and Sobolev-type inequalities that underpin the De Giorgi iteration in this non-homogeneous setting.

Abstract

In this paper we generalize the famous result of [FKS] to the double phase model. In particular, we work with minimal assumptions on the modulating coefficient by introducing a Muckenhoupt-type condition on generalized Orlicz spaces. We develop a complete theory equivalent to that of classical Muckenhoupt weights, including the boundedness of the maximal operator and Sobolev-Poincare estimates. We combine this with the De~Giorgi technique to show Hölder continuity of the solutions.

Double phase meets Muckenhoupt

TL;DR

The paper tackles Hölder regularity for double-phase variational problems with minimal regularity on the modulating coefficient by developing a Muckenhoupt-type condition in generalized Orlicz spaces. It builds a complete analytic framework, including a generalized Jensen inequality, maximal operator boundedness, density results, and Sobolev–Poincaré-type inequalities, enabling a De Giorgi–type regularity theory. Under , weak solutions are locally Hölder continuous, with explicit oscillation decay and Hölder exponent depending on , , , and the -constant. The work unifies and extends classical results (in the spirit of FKS) to non-homogeneous double-phase models, even with discontinuous modulating coefficients, advancing the regularity theory for variational problems with nonstandard growth. It also provides density and Sobolev-type inequalities that underpin the De Giorgi iteration in this non-homogeneous setting.

Abstract

In this paper we generalize the famous result of [FKS] to the double phase model. In particular, we work with minimal assumptions on the modulating coefficient by introducing a Muckenhoupt-type condition on generalized Orlicz spaces. We develop a complete theory equivalent to that of classical Muckenhoupt weights, including the boundedness of the maximal operator and Sobolev-Poincare estimates. We combine this with the De~Giorgi technique to show Hölder continuity of the solutions.
Paper Structure (14 sections, 20 theorems, 146 equations)

This paper contains 14 sections, 20 theorems, 146 equations.

Table of Contents

  1. Introduction
  2. The double phase model and function spaces
  3. The model
  4. Generalized Orlicz function spaces
  5. Muckenhoupt condition
  6. Generalized Jensen's inequality for the double phase
  7. Examples
  8. Improved generalized Jensen's inequality
  9. Boundedness of the maximal function
  10. Density of regular functions
  11. Sobolev- inequality
  12. Regularity results
  13. Caccioppoli inequality and local boundedness
  14. Hölder continuity

Key Result

theorem 1

Let $\phi(x,t)=\frac{1}{p}t^p+\frac{1}{q}a(x)t^q$, where $1<p<q$, $a:\RR^n\to [0,\infty)$, and $\phi\in\mathcal{A}$. If $u \in W^{1,\phi}(\Omega)$ is a weak solution of eq:pde, then $u$ is locally Hölder continuous. Moreover, there exists $\beta\in(0,1)$ depending on $n$, $p$, $q$ and $[\phi]_{\math for some $c>0$ depending on $n$, $p$, $q$, and $[\phi]_{\mathcal{A}}$, where $r>0$ denotes the radi

Theorems & Definitions (45)

  • theorem 1: Hölder regularity
  • lemma 1
  • proof
  • lemma 2: Norm conjugate formula
  • remark 1
  • remark 2
  • theorem 2: Generalized Jensen's inequality
  • remark 3
  • lemma 3
  • proof
  • ...and 35 more