Geometric and analytic views in existence theorems for optimal control. II. Distributed and boundary controls

Existence theorems are proved for Lagrange problems of optimization in a given domainG with possibly unbounded distributed controls inG and on the boundary ofG, and with functional relations onG and on the boundary represented by closed operators, not necessarily linear. The case where the functional relations are partial differential equations is emphasized. Recent work concerning the reduction or elimination of seminormality requirements is taken into account. Many examples are given.This research was partially supported by AFOSR Research Project 71-2122.     

Sobolev problem in the complete scale of Banach Spaces

In a bounded domainG ? ? ⁿ , whose boundary is the union of manifolds of different dimensions, we study the Sobolev problem for a properly elliptic expression of order 2m. The boundary conditions are given by linear differential expressions on manifolds of different dimensions. We study the Sobolev problem in the complete scale of Banach spaces. For this problem, we prove the theorem on a complete set of isomorphisms and indicate its applications.     

The Integral Equation Method and the Neumann Problem for the Poisson Equation on NTA Domains

The Neumann problem for the Poisson equation is studied on a general open subset G of the Euclidean space. The right-hand side is a distribution F supported on the closure of G. It is shown that a solution is the Newton potential corresponding to a distribution B ∈ε (clG), where ε(clG) is the set of all distributions with finite energy supported on the closure of G. The solution is looked for in this form and the original problem reduces to the integral equation TB = F. If the equation TB = F is solvable, then the solution is constructed by the Neumann series. The necessary and sufficient conditions for the solvability of the equation TB = F is given for NTA domains with compact boundary. The research was supported by the Academy of Sciences of the Czech Republic, Institutional Research Plan No. AV0Z10190503.     

Evolutionary weighted p-Laplacian with boundary degeneracy

The paper concerns the well-posedness problem of an evolutionary weighted p-Laplacian with boundary degeneracy. Different from the classical theory for linear equations, it is shown that the degenerate portion of the boundary should be decomposed into two parts: the strongly degenerate boundary on which the equation exhibits hyperbolic characteristics and the weakly degenerate boundary on which the equation still exhibits parabolic characteristics. We formulate reasonably the boundary value condition and establish the existence and uniqueness theorems.     

The Neumann problem for the Laplace equation on general domains

The solution of the weak Neumann problem for the Laplace equation with a distribution as a boundary condition is studied on a general open set G in the Euclidean space. It is shown that the solution of the problem is the sum of a constant and the Newtonian potential corresponding to a distribution with finite energy supported on ∂G. If we look for a solution of the problem in this form we get a bounded linear operator. Under mild assumptions on G a necessary and sufficient condition for the solvability of the problem is given and the solution is constructed. The research was supported by the Academy of Sciences of the Czech Republic, Institutional Research Plan No. AV0Z10190503.     

Fredholm composition operators on spaces of holomorphic functions

Composition operators on vector spaces of holomorphic functions are considered. Necessary conditions that range of the operator is of a finite codimension are given. As a corollary of the result it is shown that a composition operatorC on a certain Banach space of holomorphic functions on a strictly pseudoconvex domain withC ² boundary or a polydisc or a compact bordered Riemann surface or a bounded domainD such that intD = D is invertible if and only if it is a Fredholm operator if and only if is a holomorphic automorphism.     

On the Sobolev problem in the complete scale of Banach spaces

In a bounded domainG with boundary ∂G that consists of components of different dimensions, we consider an elliptic boundary-value problem in complete scales of Banach spaces. The orders of boundary expressions are arbitrary; they are pseudodifferential along ∂G. We prove the theorem on a complete set of isomorphisms and generalize its application. The results obtained are true for elliptic Sobolev problems with a parameter and parabolic Sobolev problems as well as for systems with the Douglis-Nirenberg structure. Chernigov Pedagogical Institute, Chernigov. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 9, pp. 1181–1192, September, 1999.     

The convergence of the horizontal line method for Maxwell's equations

The interior initial-boundary value problem for Maxwell's equations is considered for certain homogeneous boundary conditions and currents which satisfy corresponding compatibility conditions. Under the assumption that a solution exists, it is shown by means of suitable normed spaces that the sequence of discrete solutions obtained from the horizontal line method converges discretely to this solution. A priori estimations are given both for the discrete solutions and the solution of the given problem.     

Multisummability of formal power series solutions of partial differential equations with constant coefficients

In an earlier paper of the author's, partial differential equations with constant coefficients have been studied. Under a certain (restrictive) assumption upon the equation, those initial conditions were characterized for which the normalized formal solution of a corresponding Cauchy problem is k-summable. Here we treat the general situation and prove an analogous result, using multisummability instead of k-summability. The appropriate multisummability type is shown to depend upon the given PDE only, and can be determined from a corresponding Newton polygon.     

A free boundary problem for $\infty$–Laplace equation

We consider a free boundary problem for the p-Laplacian describing nonlinear potential flow past a convex profile K with prescribed pressure on the free stream line. The main purpose of this paper is to study the limit as of the classical solutions of the problem above, existing under certain convexity assumptions on a(x). We show, as one can expect, that the limit solves the corresponding potential flow problem for the -Laplacian in a certain weak sense, strong enough however, to guarantee uniqueness. We show also that in the special case the limit is given by the distance function. Received: 10 October 2000 / Accepted: 23 February 2001 / Published online: 19 October 2001     

Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation

In this paper, the initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation over an open bounded domain G×(0,T), G∈Rn are considered. Based on an appropriate maximum principle that is formulated and proved in the paper, too, some a priory estimates for the solution and then its uniqueness are established. To show the existence of the solution, first a formal solution is constructed using the Fourier method of the separation of the variables. The time-dependent components of the solution are given in terms of the multinomial Mittag-Leffler function. Under certain conditions, the formal solution is shown to be a generalized solution of the initial-boundary-value problem for the generalized time-fractional multi-term diffusion equation that turns out to be a classical solution under some additional conditions. Another important consequence from the maximum principle is a continuously dependence of the solution on the problem data (initial and boundary conditions and a source function) that - together with the uniqueness and existence results - makes the problem under consideration to a well-posed problem in the Hadamard sense.     

On Representation Formulas and Radiation Conditions

Solutions of linear elliptic partial differential equations in unbounded domains can be represented by boundary potentials if they satisfy certain conditions at infinity. These radiation conditions depend on the fundamental solution chosen for the integral representation. We prove some basic results about radiation conditions in a rather general framework. Fundamental solutions G are considered that are defined only on the complement of a compact set. It turns out, however, and we present examples for this, that the more interesting results only hold if G is defined on all of ℝⁿ or if it is a Green function for an exterior boundary value problem. © 1997 by B.G. Teubner Stuttgart-John Wiley & Sons, Ltd.     

Numerical solutions of the Dirichlet problem via a density theorem

Under mild conditions a certain subspace M, consisting of functions which are analytic in a simply connected domain Ω and continuous on the boundary Gamma;, is shown to have real parts which are dense, in the sup norm, in the set of all solutions to the Dirichlet problem for continuous boundary data. Similar results hold for L^p boundary data. Numerical solutions of sample Dirichlet problems are computed. © 1994 John Wiley & Sons, Inc.     

Über das Stekloffsche Eigenwertproblem: Isoperimetrische Ungleichungen für symmetrische Gebiete

Summary Isoperimetric inequalities ofPólya [6] for symmetric membranes are extended to the Stekloff problem. The given symmetric domainG _z is mapped conformally onto a circle; some (harmonic) eigenfunctions of the circle are transplanted ontoG _z ; application of Rayleigh's and Poincaré's principles to the transplanted functions gives upper bounds for a number of eigenvalues ofG _z which depends on the order of symmetry of the domain.     

Boundary regularity for a family of overdetermined problems for the Helmholtz equation

If a nonconstant solution u of the Helmholtz equation exists on a bounded domain with u satisfying overdetermined boundary conditions (u and its normal derivative both required to be constant on the boundary), then under certain assumptions the boundary of the domain is proved to be real-analytic. Under weaker assumptions, if a real-analytic portion of the boundary has a real-analytic extension, then that extension must also be part of the boundary. Also, an explicit formula for u is given and a condition (which does not involve u) is given for a bounded domain to have such a solution u defined on it. Both of these last results involve acoustic single- and double-layer potentials.     

Holomorphic Functions Having Large Images under the Action of Differential Operators

We prove in this note that, given a simply connected domainGin the complex plane and a sequence of infinite order linear differential operators generated by entire functions of subexponential type satisfying suitable conditions, then there are holomorphic functionsfonGsuch that the image of any open subset under the action of those operators onfis arbitrarily large. This generalizes an earlier result about images of derivatives. A known statement about close orbits is also strengthened.     

Harmonic maps with singular boundary value from complex hyperbolic spaces into rank one symmetric spaces

In this paper we consider the Dirichlet problem at infinity of proper harmonic maps from noncompact complex hyperbolic space to a rank one symmetric space N of noncompact type with singular boundary data . Under some conditions on f, we show that the Dirichlet problem at infinity admits a harmonic map which assumes the boundary data f continuously. Received: March 11, 1999 / Accepted April 23, 1999     

A minimum problem with free boundary in Orlicz spaces

We consider the optimization problem of minimizing in the class of functions W1,G(Ω) with , for a given φ₀?0 and bounded. W1,G(Ω) is the class of weakly differentiable functions with . The conditions on the function G allow for a different behavior at 0 and at ∞. We prove that every solution u is locally Lipschitz continuous, that it is a solution to a free boundary problem and that the free boundary, Ω∩∂{u>0}, is a regular surface. Also, we introduce the notion of weak solution to the free boundary problem solved by the minimizers and prove the Lipschitz regularity of the weak solutions and the C1,α regularity of their free boundaries near “flat” free boundary points.     

ON EXISTENCE AND UNIQUENESS IN A FREE BOUNDARY PROBLEM FROM COMBUSTION

ABSTRACT

We study a free boundary problem for the heat equation describing the propagation of laminar flames under certain geometric assumptions on the initial data. The problem arises as the limit of a singular perturbation problem, and generally no uniqueness of limit solutions can be expected. However, if the initial data is starshaped, we show that the limit solution is unique and coincides with the minimal classical supersolution. Under certain convexity assumption on the data, we prove first that the limit solution is a classical solution of the free boundary problem for a short time interval, and then that the solution, in fact, stays classical as long as it does not vanish identically.     

Existence and uniqueness of a solution for a two dimensional nonlinear inverse diffusion problem

The problem of identifying the coefficient in a square porous medium is considered. It is shown that under certain conditions of data f,g, and for a properly specified class A of admissible coefficients, there exists at least one a∈A such that (a,u) is a solution of the corresponding inverse problem.