Modulus of continuity of the solution to the Dirichlet problem for a second-order parabolic equation

The modulus of continuity of the solution to the Dirichlet problem is investigated for a second-order parabolic equation at a regular boundary point. A bound for the modulus of continuity is obtained in terms of the capacity. The coefficients of the equation are required to satisfy a Dini condition (uniformly).Translated from Matematicheskie Zametki, Vol. 19, No. 4, pp. 587–593, April, 1976.     

Modulus of continuity of a harmonic function at a boundary point

An estimate of the modulus of continuity of a harmonic function at a boundary point is found. This estimate improves a result of the author (1963), formulated in terms of Wiener series. A new condition is given for the Hölder continuity of a harmonic function at a boundary point.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 135, pp. 87–95, 1984.     

The Dirichlet problem at the Martin boundary of a fine domain

We adapt the Perron–Wiener–Brelot method of solving the Dirichlet problem at the Martin boundary of a Euclidean domain so as to cover also the Dirichlet problem at the Martin boundary of a fine domain U in ${\mathbb{R}}^n$ ($n\ge 2$) (i.e., a set U which is open and connected in the H. Cartan fine topology on ${\mathbb{R}}^n$, the coarsest topology in which all superharmonic functions are continuous). It is a complication that there is no Harnack convergence theorem for so-called finely harmonic functions. We define resolutivity of a numerical function on the Martin boundary $\mathrm{\Delta }(U)$ of U. Our main result Theorem 4.14 implies the corresponding known result for the classical case. We also obtain analogous results for the case where the upper and lower PWB-classes are defined in terms of the minimal-fine topology on the Riesz–Martin space $\stackrel{￣}{U}=U\cup \mathrm{\Delta }(U)$ instead of the natural topology. The two corresponding concepts of resolutivity are compatible.     

The integral modulus of continuity of the Dirichlet kernel and the conjugate Dirichlet kernel

Asymptotic estimates for the integral modulus of continuity of order s of the Dirichlet kernel and the conjugate Dirichlet kernel are obtained. For example, if k/2, then _s (D _k ,)=2 ^s +1/²sin ^s k/2 log(1+k/s)+O(2 ^s sin ^s k/2)holds uniformly with respect to all the parameters.Translated from Matematicheskie Zametki, Vol. 54, No. 3, pp. 98–105, September, 1993.     

Stability of the linear complementarity problem at a solution point

In this paper we study the behavior of a solution of the linear complementarity problem when data are perturbed. We give characterizations of strong stability of the linear complementarity problem at a solution. In the case of stability we give sufficient and necessary conditions.     

On the semicontinuity of the solution map to a parametric boundary control problem

This paper studies solution stability of a parametric boundary control problem governed by semilinear elliptic equation and nonconvex cost function with mixed state control constraints. Using the direct method and the first-order necessary optimality conditions, we obtain the upper semicontinuity and continuity of the solution map with respect to parameters.     

A two point boundary value problem with a rapidly oscillating solution

Summary Some numerical methods are developed for a two point boundary value problem with a rapidly oscillating solution. The two point boundary value problem is chosen to model some of the difficulties that may be expected to occur in solving the reduced wave equation at moderately high frequencies.Dedicated to lvo Babuka on his sixtieth birthdayWork supported in part by IR funds of NSWC     

Projective Dirichlet boundary condition with applications to a geometric problem

Given a domain Ω ? R~n, let λ 0 be an eigenvalue of the elliptic operator L :=Σ!(i,j)~n =1?/?xi(a~(ij0 ?/?xj) on Ω for Dirichlet condition. For a function f ∈ L~2(Ω), it is known that the linear resonance equation Lu + λu = f in Ω with Dirichlet boundary condition is not always solvable.We give a new boundary condition P_λ(u|? Ω) = g, called to be pro jective Dirichlet condition, such that the linear resonance equation always admits a unique solution u being orthogonal to all of the eigenfunctions corresponding to λ which satisfies ||u||2,2 ≤ C(||f ||_2 +|| g||_(2,2)) under suitable regularity assumptions on ?Ω and L, where C is a constant depends only on n, Ω, and L. More a priori estimates,such as W~(2,p)-estimates and the C~(2,α)-estimates etc., are given also. This boundary condition can be viewed as a generalization of the Dirichlet condition to resonance equations and shows its advantage when applying to nonlinear resonance equations. In particular, this enables us to find the new indicatrices with vanishing mean(Cartan) torsion in Minkowski geometry. It is known that the geometry of indicatries is the foundation of Finsler geometry.     

On constructing the solution to the exterior Dirichlet problem for the Helmholtz equation

In this paper a method is given for constructing the solution to the exterior Dirichlet problem for the Helmholtz equation in three dimensions. This method is modeled after the procedure of Colton and Kleinman (Proc. Roy. Soc. Edin. 86A(1980), 29–42) for solving the corresponding two-dimensional problem. The scattering problem is reformulated as an integral equation and it is shown that its solution can be represented as a convergent Neumann series for small values of the wave number. Comparisons are made between the present method and known results. Examples are given which illustrate the method.     

On the renewal of a solution of the Dirichlet boundary-value problem for a biharmonic equation

We study the problem of renewal of a solution of the Dirichlet boundary-value problem for a biharmonic equation on the basis of the known information about the boundary function. The obtained estimates of renewal error are unimprovable in certain cases. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 8, pp. 1147–1151, August, 1998.     

On the uniqueness of a solution of a two-phase free boundary problem

In this paper, we study the uniqueness problem of a two-phase elliptic free boundary problem arising from the phase transition problem subject to given boundary data. We show that in general the comparison principle between the sub- and super-solutions does not hold, and there is no uniqueness of either a viscosity solution or a minimizer of this free boundary problem by constructing counter-examples in various cases in any dimension. In one-dimension, a bifurcation phenomenon presents and the uniqueness problem has been completely analyzed. In fact, the critical case signifies the change from uniqueness to non-uniqueness of a solution of the free boundary problem. Non-uniqueness of a solution of the free boundary problem suggests different physical stationary states caused by different processes, such as melting of ice or solidification of water, even with the same prescribed boundary data. However, we prove that a uniqueness theorem is true for the initial-boundary value problem of an ε-evolutionary problem which is the smoothed two-phase parabolic free boundary problem.     

On the Dirichlet problem

A particular case of the Dirichlet problem is solved using the Convergence Theorem for discrete-time martingales and the mean value property of harmonic functions as the main tools.     

On the existence of positive solution for a kind of multi-point boundary value problem at resonance

This paper concerns the existence of positive solution for a class of second-order m-point boundary value problems under different resonant conditions. By using the Leggett-Williams norm-type theorem due to O’Regan and Zima, we obtain the existence of positive solution. An example is given to demonstrate the main results.     

A boundary modulus of continuity for a class of singular parabolic equations

Parabolic equations describing diffusion phenomena with change of phase are considered. It is demonstrated that weak solutions are continuous up to the parabolic boundary of the domain of definition. The continuity is quantitatively described by a modulus determined a priori only in terms of the data.