Related papers: Third Double-layer Potential for generalized bi-ax…
We consider a possibly multiply connected bounded open subset $\Omega$ of ${\mathbb{R}}^n$ of class $C^{\max\{1,m\},\alpha}$ for some $m\in {\mathbb{N}}$, $\alpha\in]0,1[$ and we plan to solve both the Dirichlet and the Neumann problem for…
We thoroughly analyse the double-layer potential's role in approaches to spectral sets in the spirit of Delyon--Delyon, Crouzeix and Crouzeix--Palencia. While the potential is well-studied, we aim to clarify on several of its aspects in…
We extend the method of layer potentials to manifolds with boundary and cylindrical ends. To obtain this extension along the classical lines, we have to deal with several technical difficulties due to the non-compactness of the boundary,…
This paper presents a new approach for solving the close evaluation problem in three dimensions, commonly encountered while solving linear elliptic partial differential equations via potential theory. The goal is to evaluate layer…
Physics-informed neural networks offered an alternate way to solve several differential equations that govern complicated physics. However, their success in predicting the acoustic field is limited by the vanishing-gradient problem that…
In this paper we consider an elliptic operator with constant coefficients and we estimate the maximal function of the tangential gradient of the kernel of the double layer potential with respect to its first variable. As a consequence, we…
We build a solvability theory of elliptic boundary-value problems in normed Sobolev spaces of generalized smoothness for any integrability exponent $p>1$. The smoothness is given by a number parameter and a supplementary function parameter…
In this paper, by the use of Potential Theory, some representation results for multivariate functions from the Sobolev spaces in terms of the double layer potential and the fundamental solution of Laplace's equation are pointed out.…
Using three different notions of generalized principal eigenvalue of linear second order elliptic operators in unbounded domains, we derive necessary and sufficient conditions for the validity of the maximum principle, as well as for the…
On a bounded smooth domain we study solutions of a semilinear elliptic equation with an exponential nonlinearity and a Hardy potential depending on the distance to the boundary of the domain. We derive global a priori bounds of the…
In this paper, we prove that there exists a unique weak solution to the mixed boundary value problem for a general class of semilinear second order elliptic partial differential equations with singular coefficients. Our approach is…
This is the second part of a series devoting to the generalizations and applications of common theorems in variational bifurcation theory. Using abstract theorems in the first part we obtain many new bifurcation results for quasi-linear…
Absorbing layers are sometimes required to be impractically thick in order to offer an accurate approximation of an absorbing boundary condition for the Helmholtz equation in a heterogeneous medium. It is always possible to reduce an…
The Helmholtz equation for symmetric, traceless, second-rank tensor fields in three-dimensional flat space is solved in spherical and cylindrical coordinates by separation of variables making use of the corresponding spin-weighted…
Coupled asymmetric double well ($a\phi^2-b\phi^3+c\phi^4$) one-dimensional potentials arise in the context of first order phase transitions both in condensed matter physics and field theory. Here we provide an exhaustive set of exact…
In this paper the double-sided Talor's approximations are used to obtain generalisations and improvements of some trigonometric inequalities.
In a multidimensional infinite layer bounded by two hyperplanes, the Poisson equation with the polynomial right-hand side is considered. It is shown that the Dirichlet boundary value problem and the mixed Dirichlet-Neumann boundary value…
We use the method of layer potentials to study the $R_2$ Regularity problem and the $D_2$ Dirichlet problem for second order elliptic equations of the form $\mathcal{L}u=0$, with lower order coefficients, in bounded Lipschitz domains. For…
We embed general boundary value problems for the time-harmonic Maxwell equations into the elliptic boundary value theory. This is achieved by introducing two new scalar functions to the electromagnetic field and imposing additional boundary…
We give multiplicity results for the solutions of a nonlinear elliptic equation, with an asymmetric double well potential of Van der Waals-Allen--Cahn--Hilliard type, satisfying a linear volume constraint, on a bounded Lipschitz domain…