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Let be a bounded simply connected domain in the complex plane, . Let be a neighborhood of , let be fixed, and let be a positive weak solution to the Laplace equation in Assume that has zero boundary values on in the Sobolev sense and extend to by putting on Then there exists a positive finite Borel measure on with support contained in and such thatwhenever If and if is the Green function for with pole at then the measure coincides with harmonic measure...
A second order elliptic problem with axisymmetric data is solved in a finite element space, constructed on a triangulation with curved triangles, in such a way, that the (nonhomogeneous) boundary condition is fulfilled in the sense of a penalty. On the basis of two approximate solutions, extrapolates for both the solution and the boundary flux are defined. Some a priori error estimates are derived, provided the exact solution is regular enough. The paper extends some of the results of J.T. King...
We consider the problemwhere and are smooth bounded domains in , , and We prove that if the size of the hole goes to zero and if, simultaneously, the parameter goes to zero at the appropriate rate, then the problem has a solution which blows up at the origin.
We study the existence of positive solutions for the -Laplace Emden-Fowler equation. Let and be closed subgroups of the orthogonal group such that . We denote the orbit of through by , i.e., . We prove that if for all and the first eigenvalue of the -Laplacian is large enough, then no invariant least energy solution is invariant. Here an invariant least energy solution means a solution which achieves the minimum of the Rayleigh quotient among all invariant functions. Therefore...
We consider linear elliptic equations in bounded Lipschitz domains with mixed boundary conditions on . The main feature of this boundary value problem is the appearance of both in the equation and in the boundary condition. In general we make no assumption on the sign of the coefficient . We study positivity principles and anti-maximum principles. One of our main results states that if is somewhere negative, and then there exist two eigenvalues , such the positivity principle...
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