Harmonic extension through conical surfaces

This paper establishes extension results for harmonic functions which vanish on a conical surface. These are based on a detailed analysis of expansions for the Green function of an infinite cone.

Плюригармонические определимые функции в некоторых $o$-минимальных расширениях вещественного поля

In this paper, we first try to solve the following problem: If a pluriharmonic function $f$ is definable in an arbitrary o-minimal expansion of the structure of the real field $\overline{\mathbb{R}}:=(\mathbb{R},+,-,.,0,1,<)$, does this function be locally the real part of a holomorphic function which is definable in the same expansion? In Proposition 2.1 below, we prove that this problem has a positive answer if the Weierstrass division theorem holds true for the system of the rings of real analytic definable germs at the origin of $\mathbb{R}^n$. We obtain the same answer for an o-minimal expansion of the real field which is pfaffian closed (Proposition 2.6) for the harmonic functions. In the last section, we are going to show that the Weierstrass division theorem does not hold true for the rings of germs of real analytic functions at $0\in\mathbb{R}^n$ which are definable in the o-minimal structure $(\overline{\mathbb{R}}, x^{\alpha_1},\ldots,x^{\alpha_p})$, here $\alpha_1,\ldots,\alpha_p$ are irrational real numbers.

Electrostatic images and T-matrices for simple geometries

<p>This thesis is concerned with electrostatic boundary problems and how their solutions behave depending on the chosen basis of harmonic functions and the location of the fundamental singularities of the potential. The first part deals with the method of images for simple geometries where the exact nature of the image/fundamental singularity is unknown; essentially a study of analytic continuation for Laplace's equation in 3 dimensions. For the sphere, spheroid and cylinder, new deductions are made on the location of the images of point charges and their linear or surface charge densities, by using different harmonic series solutions that reveal the image. The second part looks for analytic expressions for the T-matrix for electromagnetic scattering of simple objects in the low frequency limit. In this formalism the incident and scattered fields are expanded on an orthogonal basis such as spherical harmonics, and the T-matrix is the transformation between the coefficients of these series, providing the general solution of any electromagnetic scattering problem by a given particle at a given wavelength. For the spheroid, bispherical system and torus, the natural basis of harmonic functions for the geometry of the scatterer are used to determine T-matrix expressed in that basis, which is then transformed onto a basis of canonical spherical harmonics via the linear relationships between different bases of harmonic functions.</p>

Electrostatic images and T-matrices for simple geometries

<p>This thesis is concerned with electrostatic boundary problems and how their solutions behave depending on the chosen basis of harmonic functions and the location of the fundamental singularities of the potential. The first part deals with the method of images for simple geometries where the exact nature of the image/fundamental singularity is unknown; essentially a study of analytic continuation for Laplace's equation in 3 dimensions. For the sphere, spheroid and cylinder, new deductions are made on the location of the images of point charges and their linear or surface charge densities, by using different harmonic series solutions that reveal the image. The second part looks for analytic expressions for the T-matrix for electromagnetic scattering of simple objects in the low frequency limit. In this formalism the incident and scattered fields are expanded on an orthogonal basis such as spherical harmonics, and the T-matrix is the transformation between the coefficients of these series, providing the general solution of any electromagnetic scattering problem by a given particle at a given wavelength. For the spheroid, bispherical system and torus, the natural basis of harmonic functions for the geometry of the scatterer are used to determine T-matrix expressed in that basis, which is then transformed onto a basis of canonical spherical harmonics via the linear relationships between different bases of harmonic functions.</p>

Note on an elementary inequality and its application to the regularity of p-harmonic functions

We study the Sobolev regularity of \(p\)-harmonic functions. We show that \(|Du|^{\frac{p-2+s}{2}}Du\) belongs to the Sobolev space \(W^{1,2}_{\operatorname{loc}}\), \(s>-1-\frac{p-1}{n-1}\), for any \(p\)-harmonic function \(u\). The proof is based on an elementary inequality.

Boundary Value Problems of Thermoelasticity for Porous Sphere and for A Space with Spherical Cavity

This article is concerned with the coupled linear quasi-static theory of thermoelasticity for porous materials under local thermal equilibrium. The system of equations is based on the constitutive equations, Darcy's law of the flow of a fluid through a porous medium, Fourier's law of heat conduction, the equations of equilibrium, fluid mass conservation and heat transfer. The system of governing equations is expressed in terms of displacement vector field, the change of volume fraction of pores, the change of fluid pressure in pore network and the variation of temperature of porous material. The present paper is devoted to construct explicit solutions of the quasi-static boundary value problems (BVPs) of coupled theory of thermoelasticity for a porous elastic sphere and for a space with a spherical cavity. In this research the regular solution of the system of equations for an isotropic porous material is constructed by means of the elementary (harmonic, bi-harmonic and meta-harmonic) functions. The basic boundary value problems (the Dirichlet type boundary value problem for a sphere and the Neumann type boundary value problem for a space with a spherical cavity) are solved explicitly. The obtained solutions are given by means of the harmonic, bi-harmonic and meta-harmonic functions. For the harmonic functions the Poisson type formulas are obtained. The bi-harmonic and meta-harmonic functions are presented as absolutely and uniformly convergent series.

Structure of singular sets of some classes of subharmonic functions

In this paper, we survey the recent results on removable singular sets for the classes of $m$-subharmonic ($m-sh$) and strongly $m$-subharmonic ($sh_m$), as well as $\alpha$-subharmonic functions, which are applied to study the singular sets of $sh_{m}$ functions. In particular, for strongly $m$-subharmonic functions from the class $L_{loc}^{p}$, it is proved that a set is a removable singular set if it has zero $C_{q,s}$-capacity. The proof of this statement is based on the fact that the space of basic functions, supported on the set $D\backslash E$, is dense in the space of test functions defined in the set $D$ on the $L_{q}^{s}$-norm. Similar results in the case of classical (sub)harmonic functions were studied in the works by L. Carleson, E. Dolzhenko, M. Blanchet, S. Gardiner, J. Riihentaus, V. Shapiro, A. Sadullaev and Zh. Yarmetov, B. Abdullaev and S. Imomkulov.