Lectures on Partial Differential Equations

“…The proof can be found in [9] (see also [8,14]). The representation formula makes use of layer potentials whose definitions we now recall; see [3,9,14].…”

Section: The Direct Problemmentioning

confidence: 96%

“…In our case, we choose to use the particular symmetry of the half-space to prove the well-posedness in order to mantain a simple mathematical interpretation of the results. Therefore, we bring the problem back to an exterior domain in the whole space for which there is a vast literature on the well-posedness, see [9].…”

Section: Introductionmentioning

confidence: 99%

Asymptotic expansion for harmonic functions in the half‐space with a pressurized cavity

Aspri

Beretta

2016

Math Methods in App Sciences

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Abstract. In this paper, we address a simplified version of a problem arising from volcanology. Specifically, as reduced form of the boundary value problem for the Lamé system, we consider a Neumann problem for harmonic functions in the half-space with a cavity C. Zero normal derivative is assumed at the boundary of the half-space; differently, at ∂C, the normal derivative of the function is required to be given by an external datum g, corresponding to a pressure term exerted on the medium at ∂C. Under the assumption that the (pressurized) cavity is small with respect to the distance from the boundary of the half-space, we establish an asymptotic formula for the solution of the problem. Main ingredients are integral equation formulations of the harmonic solution of the Neumann problem and a spectral analysis of the integral operators involved in the problem. In the special case of a datum g which describes a constant pressure at ∂C, we recover a simplified representation based on a polarization tensor.Keywords. Asymptotic expansions; harmonic functions in the half-space; single and double layer potentials. AMS subject classifications. 35C20 (31B10, 35J25)

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“…One way is to find a representation formula for the solution. This kind of the solution is called a classical solution and the study could follow [27,39,25,23]. A classical solution is usually required to be k-times continuously differentiable according to the order of the PDE.…”

Section: List Of Figuresmentioning

confidence: 99%

“…So if we want to discuss a more general problem, we use the other way which generalizes the problem and discusses the properties of the solution by a variational formulation. This kind of the solution is called a weak solution and the study could follow [23,5,25]. The strategy for studying BIEs by a weak solution is exactly the same with PDEs [65,62,35].…”

Section: List Of Figuresmentioning

confidence: 99%

Boundary element approximation for Maxwell's eigenvalue problem

Wieners

Xin

2013

Math Methods in App Sciences

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We introduce a new method for computing eigenvalues of the Maxwell operator with boundary finite elements. On bounded domains with piecewise constant material coefficients, the Maxwell solution for fixed wave number can be represented by boundary integrals, which allows to reduce the eigenvalue problem to a nonlinear problem for determining the wave number along with boundary and interface traces. A Galerkin discretization yields a smooth nonlinear matrix eigenvalue problem that is solved by Newton's method or, alternatively, the contour integral method. Several numerical results including an application to the band structure computation of a photonic crystal illustrate the efficiency of this approach. Copyright © 2013 John Wiley & Sons, Ltd.

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Lectures on Partial Differential Equations

Cited by 143 publications

References 0 publications

Asymptotically Flat Initial Data¶with Prescribed Regularity at Infinity

Asymptotically Flat Initial Data¶with Prescribed Regularity at Infinity

Asymptotic expansion for harmonic functions in the half‐space with a pressurized cavity

Boundary element approximation for Maxwell's eigenvalue problem

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