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Hölder regularity for solutions of mixed boundary value problems containing boundary terms. (English) Zbl 1121.35041

Let \(\Omega\) be an open set of \(\mathbb R^n\), \(\Gamma \subset \Omega\). The author studies the bilinear form \[ a(u,v)=\int_{\Omega}[\sum_{i,j=1}^n a_{ij} u_{x_i}v_{x_j} + \sum_{i=1}^n (b_{i} u_{x_i}v+d_{i} v_{x_i}u) + cuv]\,dx + \int_{\Gamma}guv\, d\sigma \] corresponding to a mixed boundary value problem for the equation \[ a(u,v)=\int_{\Omega}( f_0 v + \sum_{i=1}^n f_{i} v_{x_i})\, dx + \int_{\Gamma}hv \,d\sigma \quad\forall v \in V=\{v \in H^1(\Omega): v=0\}. \] The coefficients of \(a(u,v)\) can be discontinuous and unbounded. The author proves the boundedness of the bilinear form and Hölder regularity for solution of the boundary value problem.

MSC:

35J25 Boundary value problems for second-order elliptic equations
35B65 Smoothness and regularity of solutions to PDEs
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