Existence, uniqueness and regularity for elliptic $p$-Laplace systems with complex coefficients

TL;DR

This work extends the theory of elliptic $p$-Laplace systems to complex-valued coefficients, establishing existence, uniqueness, and regularity results for the Dirichlet problem with a complex coefficient $a=a^R+i a^I$ under ellipticity restraints. By translating the complex problem into a real framework via decompositions and structure lemmas, the authors prove monotone-operator-based existence and energy-based uniqueness for $p\in(1,\infty)$ and derive gradient Hölder continuity when data are Hölder continuous. They also analyze the dependence of solutions on a complex-parameterized coefficient, showing differentiability along line segments in the parameter space and providing conditions under which a stronger differentiability holds. The results yield a robust mathematical foundation for complex-valued nonlinear elliptic systems and their parametric sensitivities, with potential applications to stability analyses and parameter-aware approximations in nonlinear PDE contexts.

Abstract

This paper concerns elliptic systems of $p$-Laplace type with complex valued coefficient and source term. We extend the real valued theory of the elliptic $p$-Laplace equation to the complex valued case. We establish the existence and uniqueness of solutions to the Dirichlet problem and prove the Schauder estimate in the case of Hölder continuous coefficients and source terms. We also consider families of coefficient functions parametrized by a complex variable and prove a differentiability result for the map taking the complex parameter to the corresponding solution.

Theorems & Definitions (36)

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• Definition 3.1
• Lemma 4.1
• Lemma 4.2