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View article: Elliptic free boundary transmission, the Bernoulli and the obstacle problems
Elliptic free boundary transmission, the Bernoulli and the obstacle problems Open
An elliptic boundary problem with a solution-dependent measure is studied. Applications to elliptic transmission problems with a free interface and to a generalized elliptic obstacle problem with a nonzero Neumann term on the free boundary…
View article: Generalized Burgers Equation with Measure Data
Generalized Burgers Equation with Measure Data Open
A generalized Burgers equation with measure data is studied. The existence of a weak\nsolution of an initial boundary-value problem in a bounded cylindrical domain, is\nestablished. Time-periodic solutions are shown to exist and an optimiz…
View article: On the exact controllability of a nonlinear stochastic heat equation
On the exact controllability of a nonlinear stochastic heat equation Open
The exact controllability of a nonlinear stochastic heat equation with null Dirichlet boundary conditions, nonzero initial and target values, and an interior control is established.
View article: Exact controllability for a nonlinear stochastic wave equation
Exact controllability for a nonlinear stochastic wave equation Open
The exact controllability for a semilinear stochastic wave equation with a boundary control is established. The target and initial spaces are L 2 ( G ) × H −1 ( G ) with G being a bounded open subset of R 3 and the nonlinear terms having a…
View article: Exact controllability for a semilinear wave equation with both interior and boundary controls
Exact controllability for a semilinear wave equation with both interior and boundary controls Open
The exact controllability of a semilinear wave equation in a bounded open domain of R n , with controls on a part of the boundary and in the interior, is shown. Feedback laws are established.
View article: Time‐dependent Stokes equations with measure data
Time‐dependent Stokes equations with measure data Open
We establish the existence of a unique solution of an initial boundary value problem for the nonstationary Stokes equations in a bounded fixed cylindrical domain with measure data. Feedback laws yield the source and its intensity from the …
View article: An inverse problem for evolution inclusions
An inverse problem for evolution inclusions Open
An inverse problem, the determination of the shape and a convective coefficient on a part of the boundary from partial measurements of the solution, is studied using 2‐person optimal control techniques.
View article: An inverse problem for a nonlinear Schrödingerequation
An inverse problem for a nonlinear Schrödingerequation Open
We study the dependence on the control q of the interval of definition of the solution u of the Cauchy problem ι u ′ + Δ u = − λ | u | 2 u − ι q u in ℝ 2 × (0, T ), u ( x , 0) = ω in ℝ 2 , and we prove a version of Fibich′s conjecture. Fee…
View article: Feedback laws for $N$-person quasi-variational differential games
Feedback laws for $N$-person quasi-variational differential games Open
Some optimal feedback control laws for N-person non-zero sum quasivariational di↵erential games are obtained.
View article: On strong solutions of quasi-variational inequalities
On strong solutions of quasi-variational inequalities Open
LetS be an upper semicontinuous set-valued mapping of a Hilbert space H into the closed convex subsets of H and let cp be a lower semicontinuous mapping of 7The existence of a solution of the quasi-variational inequality u E S(u) and (Au-f…
View article: On an optimal control problem for a parabolic inclusion
On an optimal control problem for a parabolic inclusion Open
Let H, U be two real Hilbert spaces and let g be a proper lower semi-continuous convex function from L 2 (0, T;H ) into R + . For each t in [0, T ], let φ(t,. ) be a proper l.s.c. convex function from H into R with effective domain Dφ(t,.)…
View article: On a steady state free boundary problem arising in combustion theory
On a steady state free boundary problem arising in combustion theory Open
The existence of a weak solution of a two-dimensional elliptic free boundary problem arising in the theory of flame propagation is shown.The model of the flame front considered is the one derived by Buckmaster.The purpose of this paper is …
View article: On a non-stationary free boundary transmission problem with continuous extraction and convection, arising in industrial processes
On a non-stationary free boundary transmission problem with continuous extraction and convection, arising in industrial processes Open
The existence of a weak solution of a non-stationary free boundary transmission problem arising in the production of industrial materials is established. The process is governed by a coupled system involving the Navier--Stokes equations an…
View article: On the initial-boundary value problem for viscous heat conducting compressible fluids
On the initial-boundary value problem for viscous heat conducting compressible fluids Open
The existence of a weak solution, local in time of an initial boundaryvalue problem for the basic system of partial differential equations of the theory of viscous, heat conducting compressible fluids is shown.Introduction.The purpose of t…
View article: On strongly nonlinear elliptic variational inequalities
On strongly nonlinear elliptic variational inequalities Open
The existence of bounded solutions of nonlinear elliptic variational inequalities is shown.The nonlinear second order elliptic operator involved has at most an exponential growth in u and a polynomial growth in Du.The regularity of the sol…
View article: Asymptotic distribution of eigenvalues and eigenfunctions of a general class of elliptic pseudo-differential operators
Asymptotic distribution of eigenvalues and eigenfunctions of a general class of elliptic pseudo-differential operators Open
The asymptotic distribution of eigenvalues and eigenfunctions of a class of elliptic pseudo-differential operators considered recently by Eskin and Visik[2], was sudied by he writer in [6].The purpose of his pper is o exend hose results o …
View article: Periodic Solutions of Non-Linear Evolution Equations in Banach Spaces
Periodic Solutions of Non-Linear Evolution Equations in Banach Spaces Open
In this paper the theory of Browder [ 2 ] and of Lions [ 3 ] on periodic solutions of non-linear evolution equations in Banach spaces is put in a more general framework so as to include the Navier-Stokes equations and their variants. An ab…
View article: On the asymptotic behavior of the spectral function of elliptic pseudo-differential operators
On the asymptotic behavior of the spectral function of elliptic pseudo-differential operators Open
Bu A TON Let ,4 be an elliptic pseudo-differential operator of order 0 on a bounded open set 2 of R with symbol/ (x, ).Let j (xj, ) be the principal pa of the symbol of in a local coordinates system and suppose that (, ) adts a iener-Hopf …
View article: Boundary value problems for elliptic convolution equations of Wiener-Hopf type in a bounded region
Boundary value problems for elliptic convolution equations of Wiener-Hopf type in a bounded region Open
Let A be an elliptic convolution operator of order a on a bounded open set G of R n , a > 0. Let Aj be the principal part of A in a local coordinates system and Aj(x j , ξ) be its symbol with a Wiener-Hopf type of factorization with respec…
View article: Non-Local Elliptic Boundary-Value Problems
Non-Local Elliptic Boundary-Value Problems Open
Let G be a bounded open set of R n with a smooth boundary ∂ G . We consider the following elliptic boundary-value problem: where A and B j are, respectively singular integro-differential operators on G and on ∂G , of orders 2m and r j with…
View article: Asymptotic distribution of eigenvalues and eigenfunctions for general linear elliptic boundary value problems
Asymptotic distribution of eigenvalues and eigenfunctions for general linear elliptic boundary value problems Open
The asymptotic distribution of eigenvalues and eigenfunctions of elliptic operators has been studied extensively by Weyl, Courant, Carleman, Pleijel and others.During the last decade, Girding [9] and Browder solved the problem for an ellip…
View article: Singular Perturbations of Non-Linear Elliptic and Parabolic Variational Boundary-Value Problems
Singular Perturbations of Non-Linear Elliptic and Parabolic Variational Boundary-Value Problems Open
Singular perturbations of linear elliptic and parabolic boundary-value problems have been studied extensively by Visik and Lyusternik (7), Huet (5), and others. It is the purpose of this paper to extend the results of (5) to the non-linear…