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The Dirichlet elliptic problem involving regional fractional Laplacian. (English) Zbl 1394.35549

Summary: In this paper, we study the solutions of elliptic equations involving regional fractional Laplacian (E) \((- \Delta)_{\Omega}^{\alpha} u = f\) in a bounded regular domain \(\Omega\) in \(\mathbb{R}^N\; (N \geq 2)\) with \(C^{2}\) boundary \(\partial\Omega\), subject to Dirichlet boundary \(g\) on \(\partial\Omega\), where \(\alpha \in (\frac{1}{2}, 1)\) and the operator \((- \Delta)_{\Omega}^{\alpha}\) denotes the regional fractional Laplacian. We prove that when \(g \equiv 0\), problem (E) admits a unique weak solution under the hypotheses that \(f \in L^{2}(\Omega),\; f \in L^{1}(\Omega,\; \rho^{\beta}dx),\; \text{and}\; f \in \mathcal{M}(\Omega, \rho^{\beta}),\; \text{where}\; \rho(x) = \mathrm{dist}(x, \partial\Omega),\; \beta = 2\alpha - 1,\; \text{and} \mathcal{M}(\Omega, \rho^{\beta})\; \text{is a space of all Radon measures}\; \nu\; \text{satisfying}\; \int_{\Omega}\rho^{\beta}d|\nu| < + \infty\). Finally, we provide an integration by parts formula for the classical solution of (E) with boundary data \(g\).{
©2018 American Institute of Physics}

MSC:

35R11 Fractional partial differential equations
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35J25 Boundary value problems for second-order elliptic equations
35D30 Weak solutions to PDEs
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
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