[Paper Review] Critical Perturbations for Second Order Elliptic Operators. Part I: Square function bounds for layer potentials

Simon Bortz, Steve Hofmann+3 more|arXiv (Cornell University)|Mar 5, 2020

Nonlinear Partial Differential Equations57 references4 citations

TL;DR

This paper establishes $L^2$ square function bounds for layer potentials associated with critical perturbations of second-order elliptic operators in divergence form, using a vector-valued $Tb$ theorem and abstract layer potential theory. The key result is uniform slice bounds for solutions, extending to operators like the generalized magnetic Schrödinger operator with small $L^n$ magnetic and $L^{n/2}$ electric potentials.

ABSTRACT

This is the first part of a series of two papers where we study perturbations of divergence form second order elliptic operators $-\mathop{\operatorname{div}} A abla$ by first and zero order terms, whose coefficients lie in critical spaces, via the method of layer potentials. In particular, we show that the $L^2$ well-posedness of the Dirichlet, Neumann and Regularity problems for complex Hermitian, block form, or constant-coefficient divergence form elliptic operators in the upper half-space are all stable under such perturbations. For instance, this allows us to claim the first results in the setting of an unbounded domain concerning the solvability of boundary value problems for the magnetic Schrödinger operator $-( abla-i{\bf a})^2+V$ when the magnetic potential ${\bf a}$ and the electric potential $V$ are accordingly small in the norm of a scale-invariant Lebesgue space. In the present paper, we establish $L^2$ control of the square function via a vector-valued $Tb$ theorem and abstract layer potentials, and use these square function bounds to obtain uniform slice bounds for solutions. The existence and uniqueness of solutions, as well as bounds for the non-tangential maximal operator, are considered in the upcoming paper.

Motivation & Objective

• To develop a framework for studying the $L^2$ Dirichlet, Neumann, and Regularity problems for second-order elliptic operators perturbed by critical lower-order terms.
• To establish square function estimates for abstract layer potential operators in the context of divergence form operators with complex, $t$-independent coefficients.
• To derive uniform slice bounds for solutions of $ \mathcal{L}u = 0$ in $\mathbb{R}^{n+1}_+$, where $\mathcal{L}$ includes critical perturbations in $L^n$ and $L^{n/2}$ spaces.
• To extend the applicability of layer potential methods to operators such as the generalized magnetic Schrödinger operator with small $L^n$ magnetic potential and $L^{n/2}$ electric potential.
• To lay the theoretical groundwork for the second paper, which will address uniqueness and solvability of boundary value problems under weak background hypotheses.

Proposed method

• Utilizes a vector-valued $Tb$ theorem to control the square function of the single layer potential in $L^2$.
• Applies abstract layer potential theory via Green's formula and jump relations to relate boundary data to solutions.
• Employs Caccioppoli-type inequalities to control gradients and higher-order $t$-derivatives of solutions on spatial slices.
• Introduces testing functions adapted to the geometry of the problem to verify the $Tb$ conditions in the vector-valued setting.
• Uses the square function bounds to derive uniform control over traces of solutions on spatial slices $\mathbb{R}^n \times \{t\}$.
• Applies Sobolev embedding and decay estimates in $Y^{1,2}$ spaces to show convergence of traces to zero at infinity under suitable decay conditions.

Experimental results

Research questions

• RQ1 Can square function estimates be established for layer potentials associated with second-order elliptic operators perturbed by complex, critical-order lower-order terms?
• RQ2 What conditions on the coefficients $B_1 \in L^n$, $B_2 \in L^n$, and $V \in L^{n/2}$ ensure the boundedness and invertibility of the associated layer potential operators in $L^2$?
• RQ3 How can square function estimates be used to derive uniform slice bounds for solutions of $\mathcal{L}u = 0$ in $\mathbb{R}^{n+1}_+$?
• RQ4 To what extent do the results extend to the generalized magnetic Schrödinger operator $-(\nabla - i\mathbf{a})A(\nabla - i\mathbf{a}) + V$ with small $\|\mathbf{a}\|_{L^n}$ and $\|V\|_{L^{n/2}}$?
• RQ5 What are the necessary and sufficient conditions on the solution $u$ for its trace on spatial slices $\mathbb{R}^n \times \{t\}$ to decay to zero at infinity?

Key findings

• The paper establishes $L^2$ square function bounds for the single layer potential via a vector-valued $Tb$ theorem, under the assumption that the perturbations $B_1, B_2 \in L^n$ and $V \in L^{n/2}$ are sufficiently small in norm.
• Uniform slice bounds for solutions $u$ of $\mathcal{L}u = 0$ are obtained, with $\|\operatorname{Tr}_t u\|_{L^p(\mathbb{R}^n)} \to 0$ as $t \to \infty$, under the condition that $\||t \nabla u\|| < \infty$.
• The estimate $\|\operatorname{Tr}_\tau u\|_{L^2(\mathbb{R}^n)} \lesssim \int_\tau^\infty \int_{\mathbb{R}^n} t |\nabla u|^2 \, dx dt$ holds uniformly in $\tau > 0$, providing an $L^2$-sup control on slices.
• The results are new for the generalized magnetic Schrödinger operator $-(\nabla - i\mathbf{a})A(\nabla - i\mathbf{a}) + V$ when $\|\mathbf{a}\|_{L^n}$ and $\|V\|_{L^{n/2}}$ are small.
• The paper proves that if $\||t \nabla u\|| < \infty$ and $u(t) \to 0$ in the sense of distributions as $t \to \infty$, then $u$ is a good $\mathcal{D}$ solution, satisfying $u_\tau \in Y^{1,2}(\mathbb{R}^{n+1}_+)$ for all $\tau > 0$.
• For solutions with $\||t \nabla \partial_t u\|| < \infty$ and $\nabla u(t) \to 0$ as $t \to \infty$, the solution is either a good $\mathcal{N}/\mathcal{R}$ solution or a constant shift of one, depending on whether $\mathcal{L}1 = 0$.