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Benjamin J. Jaye

Publications and source records attributed to Benjamin J. Jaye.

5 recordsLinked to original sources
arXiv2012

Quasilinear elliptic equations and weighted Sobolev-Poincaré inequalities with distributional weights

We introduce a class of weak solutions to the quasilinear equation $-Δ_p u = σ|u|^{p-2}u$ in an open set $Ω\subset\mathbf{R}^n$. Here $p>1$, and $Δ_p u$ is the $p$-Laplacian operator. Our notion of solution is tailored to general distributional coefficients $σ$ satisfying a certain weighted Sobolev-Poincare inequality. We also study weak solutions of the closely related equation $-Δ_p v = (p-1)|\nabla v|^p + σ$, under the same conditions on $σ$. Our results for this latter equation will allow us to characterize the class of distributions $σ$ which satisfy the Sobolev-Poincare inequality, thereby extending earlier results on the form boundedness problem for the Schrödinger operator to $p\neq 2$.

math.AP↗
arXiv2012

Existence and regularity of positive solutions to elliptic equations of Schrödinger type

We prove the existence of positive solutions with optimal local regularity to homogeneous elliptic equations of Schrödinger type, under only a form boundedness assumption on $σ\in D'(Ω)$ and ellipticity assumption on $\mathcal{A}\in L^\infty(Ω)^{n\times n}$, for an arbitrary open set $Ω\subseteq \mathbf{R}^n$. We demonstrate that there is a two way correspondence between the form boundedness and the existence of positive solutions to this equation, as well as weak solutions to certain elliptic equations with quadratic nonlinearity in the gradient. As a consequence, we obtain necessary and sufficient conditions for both the form-boundedness (with a sharp upper form bound) and the positivity of the quadratic form of the Schrödinger type operator with arbitrary distributional potential $σ\in D'(Ω)$, and give examples clarifying the relationship between these two properties.

math.AP↗
arXiv2011

Local and global behaviour of nonlinear equations with natural growth terms

This paper concerns a study of the pointwise behaviour of positive solutions to certain quasi-linear elliptic equations with natural growth terms, under minimal regularity assumptions on the underlying coefficients. Our primary results consist of optimal pointwise estimates for positive solutions of such equations in terms of two local Wolff's potentials.

math.AP↗
arXiv2011

On the Good-$λ$ inequality for nonlinear potentials

This note concerns an extension of the good-$λ$ inequality for fractional integrals, due to B. Muckenhoupt and R. Wheeden. The classical result is refined in two aspects. Firstly, general nonlinear potentials are considered; and secondly, the constant in the inequality is proven to decay exponentially. As a consequence, the exponential integrability of the gradient of solutions to certain quasilinear elliptic equations is deduced. This in turn is a consequence of certain Morrey space embeddings which extend classical results for the Riesz potential. In addition, the good-$λ$ inequality proved here provides an elementary proof of the result of Jawerth, Perez and Welland regarding the positive cone in certain weighted Triebel-Lizorkin spaces.

math.CA↗