Potential Theory in Nonlinear Elliptic Equations

Potential theory in the context of nonlinear elliptic equations extends the classical toolkit of Green’s functions and harmonic analysis to a broader class of divergence-form operators. At its core lies the search for pointwise estimates and integral representations that mimic Riesz and Wolff potentials, yet accommodate the intrinsic nonlinearity of operators such as the p-Laplace. This framework has enabled analysts to describe the precise behaviour of weak solutions, especially when the source term is a general measure or exhibits singular behaviour. Through nonlinear potentials, it is possible to establish sharp regularity results, connect local oscillations of the gradient to global function-space properties, and uncover fine boundary phenomena under minimal geometric assumptions. The theory finds rich interplay with Sobolev and Campanato spaces, yielding comprehensive estimates that bridge classical linear theory and modern geometric analysis. Its applications span models of non-Newtonian flows, electrorheological fluids and material science, underscoring the global significance of unifying potential estimates in quasilinear settings.

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Recent work has extended nonlinear potential estimates to elliptic systems with Orlicz growth, where spatially dependent divergence operators are controlled by a generalised Wolff potential. These results not only recover sharp pointwise bounds for p-Laplace systems but also open avenues for operators characterised by variable growth conditions. Complementary studies have sharpened gradient bounds for singular p-Laplace type equations driven by measure data. In these investigations, the full range of subcritical exponents is addressed and global modulus-of-continuity estimates are derived, yielding optimal regularity even in the presence of concentrated sources. Foundational advances in nonlinear potential theory have also been consolidated through unified expositions that establish broad families of potential inequalities. These syntheses demonstrate how classical analogues of fundamental solutions emerge in nonlinear regimes, offering new fine properties of solutions to measure-data problems and solidifying the conceptual framework for subsequent developments.

Potential Theory in Nonlinear Elliptic Equations publication trend

The graph below shows the total number of articles in potential theory in nonlinear elliptic equations across all publications each year (not limited to Nature Index journals).

Technical terms

p-Laplacian: A nonlinear divergence operator div(|∇u|^{p−2}∇u) generalising the Laplace operator to p-energy minimisation problems.

Nonlinear potential: An integral operator extending classical potentials (such as Riesz) to quasilinear contexts, providing pointwise solution estimates.

Wolff potential: A nonlinear potential integral that captures the influence of measure data on solutions to p-Laplace equations and related systems.

Measure data: Source terms in a differential equation represented by Radon measures rather than by functions, allowing singular or highly concentrated inputs.

Sobolev space: A function space characterised by integrability of a function and its weak derivatives, fundamental for weak solutions of PDEs.

References

1. Guide to nonlinear potential estimates. Bulletin of Mathematical Sciences (2014).
2. Wolff potentials and measure data vectorial problems with Orlicz growth. Calculus of Variations and Partial Differential Equations (2023).
3. Gradient estimates for singular $p$-Laplace type equations with measure data. Journal of the European Mathematical Society (2023).