On the nonexistence of Green's function and failure of the strong maximum principle
Luigi Orsina,Augusto C. Ponce +1 more
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In this article, it was shown that the strong maximum principle for the Schr\"odinger operator holds in each Sobolev-connected component of the set of points which cannot carry a Green's function for any Borel function.Abstract:
Given any Borel function $V : \Omega \to [0, +\infty]$ on a smooth bounded domain $\Omega \subset \mathbb{R}^{N}$, we establish that the strong maximum principle for the Schr\"odinger operator $-\Delta + V$ in $\Omega$ holds in each Sobolev-connected component of $\Omega \setminus Z$, where $Z \subset \Omega$ is the set of points which cannot carry a Green's function for $- \Delta + V$. More generally, we show that the equation $- \Delta u + V u = \mu$ has a distributional solution in $W_{0}^{1, 1}(\Omega)$ for a nonnegative finite Borel measure $\mu$ if and only if $\mu(Z) = 0$.read more
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Foundations Of Potential Theory
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Inverse Problem for a Curved Quantum Guide
Laure Cardoulis,Michel Cristofol +1 more
TL;DR: In this article, the Dirichlet Laplacian operator −∆ on a curved quantum guide in R n (n = 2, 3) with an asymptotically straight reference curve was considered, and uniqueness results for the inverse problem associated to the reconstruction of the curvature by using either observations of spectral data or a boot-strapping method were given.
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Regular points for elliptic equations with discontinuous coefficients
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Fine Regularity of Solutions of Elliptic Partial Differential Equations
Jan Malý,William P. Ziemer +1 more
TL;DR: In this paper, potential theory Quasilinear equations Fine regularity theory Variational inequalities--Regularity Existence theory References Index Notation index. But this index is not applicable to our work.
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Renormalized solutions of elliptic equations with general measure data
TL;DR: In this article, the authors studied the nonlinear monotone elliptic problem and proved the existence of a renormalized solution by an approximation procedure, where the key point is a stability result (the strong convergence in W 1,p 0 (Ω) of the truncates).
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On the nonexistence of Green's function and failure of the strong maximum principle
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