Solvability for non-smooth Schrödinger equations with singular potentials and square integrable data
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| Format: | Preprint |
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2020
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| _version_ | 1866916440959877120 |
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| author | Morris, Andrew J. Turner, Andrew J. |
| author_facet | Morris, Andrew J. Turner, Andrew J. |
| contents | We develop a holomorphic functional calculus for first-order operators $DB$ to solve boundary value problems for Schrödinger equations $-\mathrm{div}\, A \nabla u + a V u = 0$ in the upper half-space $\mathbb{R}^{n+1}_+$ with $n\in\mathbb{N}$. This relies on quadratic estimates for $DB$, which are proved for coefficients $A,a,V$ that are independent of the transversal direction to the boundary, and comprised of a complex-elliptic pair $(A,a)$ that are bounded and measurable, and a singular potential $V$ in either $L^{n/2}(\mathbb{R}^n)$ or the reverse Hölder class $B^{q}(\mathbb{R}^n)$ with $q\geq\max\{\tfrac{n}{2},2\}$. In the latter case, square function bounds are also shown to be equivalent to non-tangential maximal function bounds. This allows us to prove that the (Dirichlet) Regularity and Neumann boundary value problems with $L^2(\mathbb{R}^n)$-data are well-posed if and only if certain boundary trace operators defined by the functional calculus are isomorphisms. We prove this property when the principal coefficient matrix $A$ has either a Hermitian or block structure. More generally, the set of all complex coefficients for which the boundary value problems are well-posed is shown to be open. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2001_11901 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Solvability for non-smooth Schrödinger equations with singular potentials and square integrable data Morris, Andrew J. Turner, Andrew J. Analysis of PDEs Classical Analysis and ODEs 35J25 (Primary) 35J10, 42B37, 47D06, 47A60 (Secondary) We develop a holomorphic functional calculus for first-order operators $DB$ to solve boundary value problems for Schrödinger equations $-\mathrm{div}\, A \nabla u + a V u = 0$ in the upper half-space $\mathbb{R}^{n+1}_+$ with $n\in\mathbb{N}$. This relies on quadratic estimates for $DB$, which are proved for coefficients $A,a,V$ that are independent of the transversal direction to the boundary, and comprised of a complex-elliptic pair $(A,a)$ that are bounded and measurable, and a singular potential $V$ in either $L^{n/2}(\mathbb{R}^n)$ or the reverse Hölder class $B^{q}(\mathbb{R}^n)$ with $q\geq\max\{\tfrac{n}{2},2\}$. In the latter case, square function bounds are also shown to be equivalent to non-tangential maximal function bounds. This allows us to prove that the (Dirichlet) Regularity and Neumann boundary value problems with $L^2(\mathbb{R}^n)$-data are well-posed if and only if certain boundary trace operators defined by the functional calculus are isomorphisms. We prove this property when the principal coefficient matrix $A$ has either a Hermitian or block structure. More generally, the set of all complex coefficients for which the boundary value problems are well-posed is shown to be open. |
| title | Solvability for non-smooth Schrödinger equations with singular potentials and square integrable data |
| topic | Analysis of PDEs Classical Analysis and ODEs 35J25 (Primary) 35J10, 42B37, 47D06, 47A60 (Secondary) |
| url | https://arxiv.org/abs/2001.11901 |