Publication

Non-classical Solutions of the P-laplace Equation

2025

Abstract

In this paper we settle Iwaniec and Sbordone's 1994 conjecture concerning very weak solutions to the p-Laplace equation. Namely, on the one hand we show that distributional solutions of the p-Laplace equation in W (1;r )for p 2 and r > max {1, p - 1} are classical weak solutions if their weak derivatives belong to certain cones. On the other hand, we construct via convex integration nonenergetic distributional solutions if this cone condition is not met, thus disproving Iwaniec and Sbordone's conjecture in general.

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2025

Abstract

Official source

Related concepts (18)

Partial differential equation

In mathematics, a partial differential equation (PDE) is an equation which computes a function between various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similar to how x is thought of as an unknown number to be solved for in an algebraic equation like x2 − 3x + 2 = 0. However, it is usually impossible to write down explicit formulas for solutions of partial differential equations.

Laplace's equation

In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as or where is the Laplace operator, is the divergence operator (also symbolized "div"), is the gradient operator (also symbolized "grad"), and is a twice-differentiable real-valued function. The Laplace operator therefore maps a scalar function to another scalar function.

Distribution (mathematics)

Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative. Distributions are widely used in the theory of partial differential equations, where it may be easier to establish the existence of distributional solutions (weak solutions) than classical solutions, or where appropriate classical solutions may not exist.

Related publications (1)

Non-classical Solutions of the P-laplace Equation

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Volterra series for solving weakly non-linear partial differential equations: application to a dissipative Burger's equation