Related papers: On Neumann and Poincare problems in mathematical p…
H\"older estimates for second derivatives are proved for solutions of fully nonlinear parabolic equations in two space variables. Related techniques extend the regularity theory for fully nonlinear parabolic equations in higher dimensions.
A combination of some weighted energy estimates is applied for the Cauchy problem of quasilinear wave equations with the standard null conditions in three spatial dimensions. Alternative proofs for global solutions are shown including the…
Anisotropic elliptic equations of the second order with variable exponents in nonlinearities and the right-hand side as a diffuse measure are considered in the space $\mathbb{R}^n$. The existence of an entropy solution in anisotropic…
It is shown that analytically soluble bound states of the Schr\"odinger equation for a large class of systems relevant to atomic and molecular physics can be obtained by means of the Laplace transform of the confluent hypergeometric…
The authors propose a Nystrom method to approximate the solution of a boundary integral equation connected with the exterior Neumann problem for Laplace's equation on planar domains with corners. They prove the convergence and the stability…
It is proved that the dimension of the space of solutions of the Dirichlet problem for the harmonic functions in the unit disk with nontangential boundary limits 0 a.e. is infinite.
The article [HPS] established a monotonicity inequality for the Helmholtz equation and presented applications to shape detection and local uniqueness in inverse boundary problems. The monotonicity inequality states that if two scattering…
The Green's functions for the Laplace equation respectively satisfying the Dirichlet and Neumann boundary conditions on the upper side of an infinite plane with a circular hole are introduced and constructed. These functions enables…
We consider a class of fully non-linear parabolic equations on compact Hermitian manifolds involving symmetric functions of partial Laplacians. Under fairly general assumptions, we show the long time existence and convergence of solutions.…
Smoothness of generalized solutions for higher-order elliptic equations with nonlocal boundary conditions is studied in plane domains. Necessary and sufficient conditions upon the right-hand side of the problem and nonlocal operators under…
A class of second-order differential equations commonly arising in physics applications are considered, and their explicit hypergeometric solutions are provided. Further, the relationship with the Generalized and Universal Associated…
We consider a Neumann problem for the Laplace equation in a periodic domain. We prove that the solution depends real analytically on the shape of the domain, on the periodicity parameters, on the Neumann datum, and on its boundary integral.
In this paper we propose a scheme which allows one to find all possible exponential solutions of special class -- non-constant volume solutions -- in Lovelock gravity in arbitrary number of dimensions and with arbitrate combinations of…
We study time-dependent compactification of extra dimensions. We assume that the spacetime is spatially homogeneous, and solve the vacuum Einstein equations without cosmological constant in more than three dimensions. We consider globally…
Here a mixed problem for a nonlinear hyperbolic equation with Neumann boundary value condition is investigated, and a priori estimations for the possible solutions of the considered problem are obtained. These results demonstrate that any…
We lift the constraint of a diagonal representation of the Hamiltonian by searching for square integrable bases that support a tridiagonal matrix representation of the wave operator. Doing so results in exactly solvable problems with a…
New isoperimetric inequalities for lower order eigenvalues of the Laplacian on closed hypersurfaces, of the biharmonic Steklov problems and of the Wentzell-Laplace on bounded domains in a Euclidean space are proven. Some open questions for…
The notion of Laplace invariants is transferred to the lattices and discrete equations which are difference analogs of hyperbolic PDE's with two independent variables. The sequence of Laplace invariants satisfy the discrete analog of…
Anisotropic exponential cosmological solutions for a space of arbitrary dimension filled with ordinary matter in the 4th and 5th orders of Lovelock gravity are obtained. Also we have supposed a generalization of such solutions on an…
In this article, we study the existence and multiplicity of solutions of the following $(p,q)$-Laplace equation with singular nonlinearity: \begin{equation*} \left\{\begin{array}{rllll} -\Delta_{p}u-\ba\Delta_{q}u & = \la u^{-\de}+ u^{r-1},…