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Laplace operator with δ-like potentials
Russian Mathematics, 2014Let \(\Omega \) be a simply connected domain with smooth boundary in \(\mathbb R^2\), and \(M=[x_0,y_0]\) be a fixed inner point of the domain \(\Omega \). Denote \(\Omega_0 =\Omega \setminus \{ M\} \). Denote by \(G(x,y,x_0,y_0)\) the Green function of the Dirichlet problem for the Laplace equation corresponding to \(\Omega \) and \([x_0,y_0]\).
Kanguzhin, B. E. +2 more
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2017
We consider what is perhaps the most important of all partial differential operators, theLaplace operator (Laplacian) on \(\mathbb {R}^n\).
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We consider what is perhaps the most important of all partial differential operators, theLaplace operator (Laplacian) on \(\mathbb {R}^n\).
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Laplace-Radon Integral Operators
1994Laplace-Radon integral operators studied in this chapter were designed to supply asymptotic solutions (w.r.t. differentability) to differential equations on complex manifolds. Prior to considering these operators in general let us describe the representation of singular solutions to differential equations we intend to use in our constructions.
Boris Sternin, Victor Shatalov
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MULTIPLE EIGENVALUES OF THE LAPLACE OPERATOR
Mathematics of the USSR-Sbornik, 1988The estimates from above for the multiplicity of the eigenvalues of the Schrödinger operators on compact Riemannian two-dimensional manifolds are obtained. The paper contains also several examples illustrating the sharpness of the results.
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Two Theorems Concerning the Laplace Operator
American Journal of Physics, 1963A theorem is derived which gives the mean value of a well-behaved function in a small spherical volume in terms of the value of the function and its Laplacian at the center of the sphere. The magnitude of ψ(x,y,z) at the center exceeds the mean value at neighboring points or falls short of the mean value according to whether −∇2ψ is positive or ...
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