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Regularizing effects concerning elliptic equations with a superlinear gradient term

Marta Latorre Balado, Martina Magliocca, Sergio Segura de León

TL;DR

The paper analyzes a homogeneous Dirichlet elliptic problem with gradient growth of the form $g(u)|\nabla u|^q$, $1<q<2$, under $L^m(\Omega)$ or measure data. It identifies the optimal far-field decay $|s|^\alpha g(s)$ through $\alpha=\frac{N(q-1)-mq}{N-2m}$ (for $m>1$) that guarantees existence, and classifies the gradient-growth into sublinear, linear, and superlinear regimes with corresponding existence results in finite-energy or renormalized senses. The authors develop a refined a priori estimate framework and compactness arguments, including a Marcinkiewicz-based approach for measure data, to obtain convergence and higher regularity in various parameter ranges, showing consistency with GMP (constant $g$) and PS (quadratic growth) in the appropriate limits. The work demonstrates a regularizing effect of the gradient term on existence and summability, extends the theory to subquadratic growth and discontinuous coefficients, and provides a unified treatment across data regimes from Lebesgue to measure data. This advances the understanding of nonlinear elliptic equations with gradient-driven nonlinearities and their solvability under low-regularity data.

Abstract

We consider the homogeneous Dirichlet problem for an elliptic equation driven by a linear operator with discontinuous coefficients and having a subquadratic gradient term. This gradient term behaves as $g(u)|\nabla u|^q$, where $1<q<2$ and $g(s)$ is a continuous function. Data belong to $L^m(Ω)$ with $1\le m <\frac{N}{2}$ as well as measure data instead of $L^1$-data, so that unbounded solutions are expected. Our aim is, given $1\le m<\frac N2$ and $1<q<2$, to find the suitable behaviour of $g$ close to infinity which leads to existence for our problem. We show that the presence of $g$ has a regularizing effect in the existence and summability of the solution. Moreover, our results adjust with continuity with known results when either $g(s)$ is constant or $q=2$.

Regularizing effects concerning elliptic equations with a superlinear gradient term

TL;DR

The paper analyzes a homogeneous Dirichlet elliptic problem with gradient growth of the form , , under or measure data. It identifies the optimal far-field decay through (for ) that guarantees existence, and classifies the gradient-growth into sublinear, linear, and superlinear regimes with corresponding existence results in finite-energy or renormalized senses. The authors develop a refined a priori estimate framework and compactness arguments, including a Marcinkiewicz-based approach for measure data, to obtain convergence and higher regularity in various parameter ranges, showing consistency with GMP (constant ) and PS (quadratic growth) in the appropriate limits. The work demonstrates a regularizing effect of the gradient term on existence and summability, extends the theory to subquadratic growth and discontinuous coefficients, and provides a unified treatment across data regimes from Lebesgue to measure data. This advances the understanding of nonlinear elliptic equations with gradient-driven nonlinearities and their solvability under low-regularity data.

Abstract

We consider the homogeneous Dirichlet problem for an elliptic equation driven by a linear operator with discontinuous coefficients and having a subquadratic gradient term. This gradient term behaves as , where and is a continuous function. Data belong to with as well as measure data instead of -data, so that unbounded solutions are expected. Our aim is, given and , to find the suitable behaviour of close to infinity which leads to existence for our problem. We show that the presence of has a regularizing effect in the existence and summability of the solution. Moreover, our results adjust with continuity with known results when either is constant or .

Paper Structure

This paper contains 25 sections, 27 theorems, 293 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Our results
  4. Plan of the paper
  5. Assumptions and Statement of results
  6. Notation
  7. Assumptions
  8. Notions of solution
  9. Main results
  10. Connection among parameters
  11. Our starting point
  12. A priori estimates
  13. Finite energy solutions
  14. Renormalized solutions with $L^m(\Omega)$ data
  15. Renormalized solutions with measure data
  16. ...and 10 more sections

Key Result

Theorem 2.7

Using the above notation, assume that $\|f\|_{L^m(\Omega)}$ is small enough. Furthermore, assuming a source $\mu\in\mathcal{M}_b(\Omega)$ with $\|\mu\|_{\mathcal{M}_b}$ small enough, if $\frac{N(q-1)-q}{N-2}< \alpha<q-1$, then there exists a renormalized solution to problem_measure.

Figures (1)

  • Figure :

Theorems & Definitions (49)

  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Theorem 2.7: Existence results in the superlinear case
  • Theorem 2.8: Existence results in the linear case
  • Proposition 2.9
  • Remark 2.10
  • ...and 39 more