Solvability of boundary value problem for Schrödinger Equations with Reverse Hölder Potentials on $L^p$ and endpoint spaces

Abstract: In this paper we discuss the solvability of the Neumann and Regularity boundary value problem of elliptic Schrödinger-type equation $-\DIV(A(x)\nabla u(x,t))+V(x)u(x,t)=0$ with bounded measurable uniformly elliptic coefficinets $A(x)$ independent of $t$ and $V$ in Reverse Hölder class $\mathcal{B}<em>q$, and Neumann boundary data $\partial</em>{ν<em>A}u(x,0)=f(x)\in H^p</em>{\mathcal{L}}(\rn)$, or Regularity data $u(x,0)=g\in H<sup>{1,p}_V(\rn)$, utilizing the method of layer potential. We prove the solvability when $A$ is a small $L<sup>\infty$ perturbation of a matrix satisfying De Giorgi-Nash-Moser bounds. Besides we also give the Campanato norm estimate of the double layer potential related to the Dirichlet problem with boundary data in certain Campanato-type spaces.

• The paper proves well-posedness of Neumann and Regularity boundary value problems for Schrödinger equations using layer potential methods in endpoint spaces.
• It introduces a novel molecular decomposition for Hardy spaces and applies sharp Rellich-type estimates to handle reverse Hölder potentials and nonsmooth coefficients.
• The work extends classical L2 results to Lp and Hardy–Sobolev spaces, providing robust analytic tools for boundary trace theory in complex settings.

Solvability of Boundary Value Problems for Schrödinger Equations with Reverse Hölder Potentials on $L^p$ and Endpoint Spaces

Introduction and Problem Formulation

This paper addresses the Neumann and Regularity problems for second-order divergence-form Schrödinger-type elliptic equations with singular lower-order potentials. Specifically, the equation is

$$\mathcal{L}u = -\mathrm{div}(A(x)\nabla u(x, t)) + V(x)u(x, t) = 0,$$

posed on the upper half-space, with $A(x)$ a bounded, uniformly elliptic, possibly non-symmetric and $t$-independent coefficient matrix, and $V\in\mathcal{B}_q$ is a nonnegative potential satisfying a reverse Hölder condition with $q > n/2$. Neumann and Regularity data belong to Hardy-type endpoint function spaces intricately adapted to the Schrödinger operator, namely $H^p_\mathcal{L}$ and $H^{1,p}_V$.

The solvability of boundary value problems is established using layer potential methods under optimal smallness assumptions on perturbations of $A$ from reference operators that satisfy quantitative De Giorgi-Nash-Moser continuity. The main innovation is the treatment of endpoint function spaces, new molecular decompositions for $H^p_\mathcal{L}$ for $$\mathcal{L}u = -\mathrm{div}(A(x)\nabla u(x, t)) + V(x)u(x, t) = 0,$$0 near 1, and the verification of Rellich-type boundary inequalities in this nonsmooth, lower-order context.

A Priori Analysis: Reverse Hölder Potentials and Functional Calculus

The technical core is the quantitative exploitation of the reverse Hölder condition for $$\mathcal{L}u = -\mathrm{div}(A(x)\nabla u(x, t)) + V(x)u(x, t) = 0,$$1. This structure yields:

• Local maximal function characterizations and robust Fefferman–Phong inequalities controlling the weighted $$\mathcal{L}u = -\mathrm{div}(A(x)\nabla u(x, t)) + V(x)u(x, t) = 0,$$2 norm of $$\mathcal{L}u = -\mathrm{div}(A(x)\nabla u(x, t)) + V(x)u(x, t) = 0,$$3 by energy,
• Efficient control of the fundamental solution, Green’s function, and their derivatives, with polynomial spatial decay, regularity in the boundary variable, and explicit dependence on the local "scaling function" $$\mathcal{L}u = -\mathrm{div}(A(x)\nabla u(x, t)) + V(x)u(x, t) = 0,$$4 associated to $$\mathcal{L}u = -\mathrm{div}(A(x)\nabla u(x, t)) + V(x)u(x, t) = 0,$$5,
• Equivalence classes of endpoint Hardy and Campanato-type spaces, crucial for handling nonvanishing mean oscillation and noncancellation in the boundary data.

The estimates also justify passage to singular integral operator theory for boundary layer potentials, instrumental in reducing boundary value problems to invertibility questions for associated boundary operators.

Function Spaces and Molecular Decomposition

For $$\mathcal{L}u = -\mathrm{div}(A(x)\nabla u(x, t)) + V(x)u(x, t) = 0,$$6 near 1, the analysis operates in atomic or molecular variants of Hardy spaces tailored to the operator—$$\mathcal{L}u = -\mathrm{div}(A(x)\nabla u(x, t)) + V(x)u(x, t) = 0,$$7—where atoms and molecules are allowed noncancellation in a controlled regime depending on the local behavior of $$\mathcal{L}u = -\mathrm{div}(A(x)\nabla u(x, t)) + V(x)u(x, t) = 0,$$8. Key contributions in this layout include:

• Proof of a new molecular decomposition tailored to the Schrödinger context for $$\mathcal{L}u = -\mathrm{div}(A(x)\nabla u(x, t)) + V(x)u(x, t) = 0,$$9, robust down to the sharp range $A(x)$0,
• Equivalent characterizations via local maximal functions and identification of the endpoint Hardy–Sobolev space $A(x)$1, underpinning the Regularity problem formulation,
• Establishment of precise embedding and duality relations that allow the reduction of $A(x)$2-type boundary regularity questions to control of layer potentials.

Such analyses rely heavily on the quantitative decay properties of the underlying fundamental solution, as controlled by the reverse Hölder assumption.

Layer Potentials: Estimates and Invertibility Regime

The central analytic device is the use of single and double layer potentials. The fundamental results proved are:

• $A(x)$3 and $A(x)$4 maximal function bounds for single and double layer potentials, covering $A(x)$5 for some $A(x)$6,
• Campanato/Hölder estimates for the double layer potentials for Dirichlet data in $A(x)$7,
• Uniform invertibility of the conormal derivative and single-layer potentials—$A(x)$8 and $A(x)$9—acting between $t$0, $t$1, and related endpoint spaces, provided the elliptic part is a sufficiently small $t$2 perturbation of a block or real symmetric matrix satisfying DG-NM bounds.

Of note is the treatment of endpoint regularity where the lack of cancellation in the boundary values (due to the presence of $t$3) is a genuine technical obstacle, here resolved via detailed Rellich inequalities and pointwise kernel bounds.

Main Theorem and Consequences

The main solvability result asserts unique, layer-potential-based solutions for both Neumann and Regularity problems in the above setting, with data in $t$4 and $t$5 for the optimal range $t$6 (or to $t$7 for $t$8 Regularity). In particular, for $t$9 real symmetric (or block), or small $V\in\mathcal{B}_q$0 perturbations thereof,

• Both $V\in\mathcal{B}_q$1 and $V\in\mathcal{B}_q$2 are solvable by the method of layer potentials in $V\in\mathcal{B}_q$3 and $V\in\mathcal{B}_q$4 for $V\in\mathcal{B}_q$5 slightly below and above 1,
• Solutions possess nontangential limits, and the traces and conormal derivatives are controlled in the appropriate endpoint spaces,
• Unique solvability holds for functions with bounded energy and nontangential maximal function norm.

These results strictly generalize the established $V\in\mathcal{B}_q$6 outcomes (e.g., [Morris, Turner, "Solvability for non-smooth Schrödinger equations", JFA 2025]), and provide the first full account for endpoint Hardy, Hardy-Sobolev, and Campanato spaces for elliptic Schrödinger equations with rough coefficients and reverse Hölder potentials.

Implications and Directions

The analysis clarifies the delicate role played by nontrivial lower-order structure in elliptic and parabolic PDEs, notably in the failure of classical cancellation in boundary traces, and provides a rigorous functional framework for their well-posedness. Methods developed here suggest new directions:

• Extension of layer potential methods for nonhomogeneous equations with complex or oscillation-dominated lower-order terms,
• Refinements for Schrödinger-type systems with Carleson or more singular potentials,
• Further exploration of molecular Hardy spaces for other singular integral operators, and quantitative boundary regularity for nondivergence form operators.

The techniques established provide a foundation for the non-tangential boundary theory of other relevant PDEs with analytic, geometric, or probabilistic data.

Conclusion

This work systematically establishes the well-posedness and trace theory for Neumann and Regularity boundary value problems for elliptic Schrödinger operators with rough coefficients and reverse Hölder potentials on $V\in\mathcal{B}_q$7 and endpoint (Hardy, Hardy–Sobolev, Campanato) spaces. Central novelties include the endpoint molecular decomposition, sharp Rellich-type estimates, and invertibility regimes for singular integral boundary operators in these spaces, enabling a robust boundary layer potential solution theory beyond classical settings. The results and methods contribute substantially to the understanding and analysis of non-self-adjoint, nonsmooth, lower-order elliptic boundary value problems.