Perturbation of a Two-Point Problem

We investigate the problem of the effect of integral terms in boundary conditions on the well-posedness of nonlocal boundary-value problems for partial differential equations.     

Nonlocal Boundary-Value Problem for a Second-Order Partial Differential Equation in an Unbounded Strip

Ukrainian Mathematical Journal - The conditions of well-posedness of nonlocal boundary-value problems are established for a second-order linear partial differential equation in an unbounded strip...     

DC Optimization Approach to Metric Regularity of Convex Multifunctions with Applications to Infinite Systems

The paper develops a new approach to the study of metric regularity and related well-posedness properties of convex set-valued mappings between general Banach spaces by reducing them to unconstrained minimization problems with objectives given as the difference of convex (DC) functions. In this way, we establish new formulas for calculating the exact regularity bound of closed and convex multifunctions and apply them to deriving explicit conditions ensuring well-posedness of infinite convex systems described by inequality and equality constraints.     

The impedance boundary-value problem of diffraction by a strip

We consider the impedance boundary-value problem for the Helmholtz equation originated by the problem of wave diffraction by an infinite strip with imperfect conductivity. The two possible different situations of real and complex wave numbers are considered. Bessel potential spaces are used to deal with the problem, and the identification of corresponding operators of single and double layer potentials allow a reformulation of the problem into a system of integral equations. The well-posedness of the problem is obtained for a set of impedance parameters (and wave numbers), after the incorporation of some compatibility conditions on the data. At the end, an improvement of the regularity of the solution is derived for the same set of parameters previously considered.     

Problems for equations with special parabolic operator of fractional differentiation

We establish the well-posedness of the Cauchy problem and the two-point boundary-value problem for an equation with an operator of fractional differentiation that corresponds to the singular parabolic Beltrami – Laplace operator on a surface of the Dini class.     

On general boundary-value problems for nonlinear elliptic equations of second order in a multiply connected plane domain

This paper deals with some general irregular oblique derivative problems for nonlinear uniformly elliptic equations of second order in a multiply connected plane domain. Firstly, we state the well-posedness of a new set of modified boundary conditions. Secondly, we verify the existence of solutions of the modified boundary-value problem for harmonic functions, and then prove the solvability of the modified problem for nonlinear elliptic equations, which includes the original boundary-value problem (i.e. boundary conditions without involving undertermined functions data). Here, mainly, the location of the zeros of analytic functions, a priori estimates for solutions and the continuity method are used in deriving all these results. Furthermore, the present approach and setting seems to be new and different from what has been employed before.The research was partially supported by a UPGC Grant of Hong Kong.     

Nonlocal two-point boundary-value problems in a layer with differential operators in the boundary condition

We obtain criteria of well-posedness and strong well-posedness (smoothing of solutions as compared with given functions) of boundary-value problems for linear partial differential evolution equations in an infinite layer. The boundary condition is nonlocal and gives a relation between the values of the unknown function and its derivatives with respect to spatial coordinates on shifts of connected components of the boundary of the layer inside the layer.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 8, pp. 1122–1128, August, 1995.     

On the optimal control of the Schlögl-model

Optimal control problems for a class of 1D semilinear parabolic equations with cubic nonlinearity are considered. This class is also known as the Schlögl model. Main emphasis is laid on the control of traveling wave fronts that appear as typical solutions to the state equation. The well-posedness of the optimal control problem and the regularity of its solution are proved. First-order necessary optimality conditions are established by standard adjoint calculus. The state equation is solved by the implicit Euler method in time and a finite element technique with respect to the spatial variable. Moreover, model reduction by Proper Orthogonal Decomposition is applied and compared with the numerical solution of the full problem. To solve the optimal control problems numerically, the performance of different versions of the nonlinear conjugate gradient method is studied. Various numerical examples demonstrate the capacities and limits of optimal control methods.     

Well-posedness of initial value problems for singular parabolic equations

We study well-posedness of initial value problems for a class of singular quasilinear parabolic equations in one space dimension. Simple conditions for well-posedness in the space of bounded nonnegative solutions are given, which involve boundedness of solutions of some related linear stationary problems. By a suitable change of unknown, the above results can be applied to classical initial-boundary value problems for parabolic equations with singular coefficients, as the heat equation with inverse square potential.     

Local and nonlocal boundary-value problems for third-order hyperbolic equations

This paper studies the problems of existence of classical solutions to the Goursat and Dirichlet problems and also to some nonlocal boundary-value problems for a linear third-order hyperbolic equation in a rectangular domain. The problems studied are reduced to a uniquely solvable integral equation. Thus, theorems of existence, uniqueness, and stability of solutions of the problem are proved. The effects of the influence of coefficients standing by lower derivatives on the well-posedness of the problems studied are revealed.     

Representation and investigation of solutions of a nonlocal boundary-value problem for a system of partial differential equations

We study the boundary-value problem for a system of partial differential equations with constant coefficients with conditions nonlocal in time. By using a metric approach, we prove the well-posedness of the problem in the scale of Sobolev spaces of functions periodic in space variables. By using matrix calculus, we construct an explicit representation of a solution.     

Some boundary-value problems for a class of fourth-order differential equations on a graph

We study the problem of nondegeneracy of some boundary-value problems for a class of fourth-order differential equations on a geometric graph (topological net), present conditions for the existence of the Green function for boundary-value problems under consideration, and analyze its principal properties.     

On viscous incompressible flows of nonsymmetric fluids with mixed boundary conditions

In this paper we are concerned with the initial boundary value problem for the micropolar fluid system in nonsmooth domains with mixed boundary conditions. The considered boundary conditions are of two types: Navier’s slip conditions on solid surfaces and Neumann-type boundary conditions on free surfaces. The Dirichlet boundary condition for the microrotation of the fluid is commonly used in practice. However, the well-posedness of problems with different types of boundary conditions for microrotation are completely unexplored. The present paper is devoted to the proof of the existence, regularity and uniqueness of the solution in distribution spaces.     

A trace regularity result for thermoelastic equations with application to optimal boundary control

We consider a mixed problem for a Kirchoff thermoelastic plate model with clamped boundary conditions. We establish a sharp regularity result for the outer normal derivative of the thermal velocity on the boundary. The proof, based upon interpolation techniques, benefits from the exceptional regularity of traces of solutions to the elastic Kirchoff equation. This result, which complements recent results obtained by the second and third authors, is critical in the study of optimal control problems associated with the thermoelastic system when subject to thermal boundary control. Indeed, the present regularity estimate can be interpreted as a suitable control-theoretic property of the corresponding abstract dynamics, which is crucial to guarantee well-posedness for the associated differential Riccati equations.     

Scalarization for pointwise well-posed vectorial problems

The aim of this paper is to develop a method of study of Tykhonov well-posedness notions for vector valued problems using a class of scalar problems. Having a vectorial problem, the scalarization technique we use allows us to construct a class of scalar problems whose well-posedness properties are equivalent with the most known well-posedness properties of the original problem. Then a well-posedness property of a quasiconvex level-closed problem is derived.      

Well-posedness of non-autonomous linear evolution equations for generators whose commutators are scalar

We prove the well-posedness of non-autonomous linear evolution equations for generators \({A(t): D(A(t)) \subset X \to X}\) whose pairwise commutators are complex scalars, and in addition, we establish an explicit representation formula for the evolution. We also prove well-posedness in the more general case where instead of the onefold commutators only the p-fold commutators of the operators A(t) are complex scalars. All these results are furnished with rather mild stability and regularity assumptions: Indeed, stability in X and strong continuity conditions are sufficient. Additionally, we improve a well-posedness result of Kato for group generators A(t) by showing that the original norm continuity condition can be relaxed to strong continuity. Applications include Segal field operators and Schrödinger operators for particles in external electric fields.     

The residual velocity method applied to a steady free boundary-value problem of vector Laplacian type

We consider a free boundary-value problem based on a simplifiedmodel of two-phase flow in porous media. The model has two independentvariables on each side of the free interface. At the interfaceat steady state, five mixed Dirichlet and Neumann conditionsare given. The movement of the interface in time-dependent situationscan be reduced to a normal motion proportional to the residualin one of the steady-state interface conditions (the ellipticinterior problems and the other interface conditions are satisfiedat each time). Following previous work, we consider the useof other residuals for the normal velocity that have superiornumerical properties. The well-posedness criteria for this vectorexample are particularly clear. The advantages of the correctlychosen, non-physical residual velocities are demonstrated innumerical computations. Although the finite-difference implementationin this work is not applicable to general problems, it has superiorperformance to previous implementations.     

Regularity of solutions of initial-boundary value problems for parabolic equations in domains with conical points

The purpose of this paper is to establish the well-posedness and the regularity of solutions of the initial-boundary value problems for general higher order parabolic equations in infinite cylinders with the bases containing conical points.     

The Cauchy Problem for the Radiation Transfer Equation with Fresnel and Lambert Matching Conditions

Mathematical Notes - The well-posedness of the initial boundary-value problem for the nonstationary radiation transfer equation in a three-dimensional bounded domain with generalized matching...     

Conditions of regularity of a general differential boundary-value problem for improperly elliptic equations

The Fredholm property and well-posedness of a general differential boundary-value problem for a general improperly elliptic equation are analyzed in a two-dimensional bounded domain with smooth boundary.