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# Dirichlet boundary condition

About: Dirichlet boundary condition is a research topic. Over the lifetime, 11722 publications have been published within this topic receiving 199137 citations.

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TL;DR: In this article, the square root of the Laplacian (−△) 1/2 operator was obtained from the harmonic extension problem to the upper half space as the operator that maps the Dirichlet boundary condition to the Neumann condition.

Abstract: The operator square root of the Laplacian (−△) 1/2 can be obtained from the harmonic extension problem to the upper half space as the operator that maps the Dirichlet boundary condition to the Neumann condition. In this paper we obtain similar characterizations for general fractional powers of the Laplacian and other integro-differential operators. From those characterizations we derive some properties of these integro-differential equations from purely local arguments in the extension problems.

2,696 citations

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TL;DR: In this article, a Sommerfeld radiation condition (2.2) was proposed for problems requiring a prescribed open boundary, and two severe tests were used to demonstrate the applicability of the open boundary condition: collapsing bubble, a dynamic event which excites many different internal gravity waves.

Abstract: A Sommerfeld radiation condition (2.2) is proposed for problems requiring a prescribed open boundary. The equations must be hyperbolic in nature (although the author believes that they may also be good for some elliptic and parabolic problems). It is proven that the proposed condition was shown to be free of reflection for single wave propagation. Two severe tests were used to demonstrate the applicability of the open boundary condition: (i) the collapsing bubble, a dynamic event which excites many different internal gravity waves. The results show minimum distortion. (ii) The spatially growing K-H instability. This test differs from the previous one in that the only waves excited are those corresponding to the maximum unstable wavelengths. In this case, the maximum amplitude is reached at the open boundary. As it has been shown, the open boundary condition (2.2) produces minimum distortion.

1,870 citations

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TL;DR: In this paper, the single smooth coefficient of the elliptic operator LY = v yv can be determined from knowledge of its Dirichlet integrals for arbitrary boundary values on a fixed region 2 C R', n? 3.

Abstract: In this paper, we show that the single smooth coefficient of the elliptic operator LY = v yv can be determined from knowledge of its Dirichlet integrals for arbitrary boundary values on a fixed region 2 C R', n ? 3. From a physical point of view, we show that an isotropic conductivity can be determined by steady state measurements at the boundary.

1,608 citations

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TL;DR: An algorithm is described for rapid solution of classical boundary value problems (Dirichlet an Neumann) for the Laplace equation based on iteratively solving integral equations of potential theory using CPUs proportional to n.

Abstract: An algorithm is described for rapid solution of classical boundary value problems (Dirichlet an Neumann) for the Laplace equation based on iteratively solving integral equations of potential theory. CPU time requirements for previously published algorithms of this type are proportional to n2, where n is the number of nodes in the discretization of the boundary of the region. The CPU time requirements for the algorithm of the present paper are proportional to n, making it considerably more practical for large scale problems.

1,426 citations

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TL;DR: In this article, the uniqueness of the positive, radially symmetric solution to the differential equation Δu−u+up=0 (with p>1) in a bounded or unbounded annular region in Rn for all n ≥ 1, with the Neumann boundary condition on the inner ball and the Dirichlet boundary condition decaying to zero in the case of an unbounded region, was established.

Abstract: We establish the uniqueness of the positive, radially symmetric solution to the differential equation Δu−u+up=0 (with p>1) in a bounded or unbounded annular region in Rn for all n≧1, with the Neumann boundary condition on the inner ball and the Dirichlet boundary condition on the outer ball (to be interpreted as decaying to zero in the case of an unbounded region). The regions we are interested in include, in particular, the cases of a ball, the exterior of a ball, and the whole space. For p=3 and n=3, this a well-known result of Coffman, which was later extended by McLeod & Serrin to general n and all values of p below a certain bound depending on n. Our result shows that such a bound on p is not needed. The basic approach used in this work is that of Coffman, but several of the principal steps in the proof are carried out with the help of Sturm's oscillation theory for linear second-order differential equations. Elementary topological arguments are widely used in the study.

1,338 citations