Optimal control of the obstacle in semilinear variational inequalities

We consider an optimal control problem where the state satisfies an obstacle type semilinear variational inequality and the control function is the obstacle. The state is chosen to be close to a desired profile while the obstacle is not too large in H ₀ ¹ (), and H ²-bounded. We prove that an optimal control exists and give necessary optimality conditions, using approximation techniques.     

Optimal Control of the Obstacle for a Parabolic Variational Inequality

An optimal control problem for a parabolic obstacle variational inequality is considered. The obstacle in L²(0, T; H²(Ω) ∩ H¹₀(Ω)) with ψ_t ∈ L²(Q) is taken as the control, and the solution to the obstacle problem is taken as the state. The goal is to find the optimal control so that the state is close to the desired profile while the norm of the obstacle is not too large. Existence and necessary conditions for the optimal control are established.     

Optimization of the first Steklov eigenvalue in domains with holes: a shape derivative approach

The best Sobolev trace constant is given by the first eigenvalue of a Steklov-like problem. We deal with minimizers of the Rayleigh quotient ‖u‖² _H ¹ _(Ω) ²/‖u‖² _L ² _(∂Ω) for functions that vanish in a subset A⊂ Ω, which we call the hole. We look for holes that minimize the best Sobolev trace constant among subsets of Ω with prescribed volume. First, we find a formula for the first variation of the first eigenvalue with respect to the hole. As a consequence of this formula, we prove that when Ω is a ball the symmetric hole (a centered ball) is critical when we consider deformations that preserves volume but is not optimal. Finally, we prove that by the Finite Element Method we can approximate the optimal configuration and, by means of the shape derivative, we design an algorithm to compute the discrete optimal holes. Mathematics Subject Classification (2000) 35P15, 49K20, 49M25, 49Q10     

Identification of the Time Derivative Coefficient in a Fast Diffusion Degenerate Equation

In this paper, we deal with the identification of the space variable time derivative coefficient u in a degenerate fast diffusion differential inclusion. The function u is vanishing on a subset strictly included in the space domain Ω. This problem is approached as a control problem (P) with the control u. An approximating control problem (P _ε ) is introduced and the existence of an optimal pair is proved. Under certain assumptions on the initial data, the control is found in W ^2,m (Ω), with m>N, in an implicit variational form. Next, it is shown that a sequence of optimal pairs (u_e^*,y_e^*)(u_{\varepsilon }^{\ast },y_{\varepsilon }^{\ast }) of (P _ε ) converges as ε goes to 0 to a pair (u ^*,y ^*) which realizes the minimum in (P), and y ^* is the solution to the original state system.     

On the regularity of solutions of boundary-value problem for quasilinear elliptic systems with quadratic nonlinearity

The partial regularity up to the boundary of a domain is established for a solution u ∈ H¹ (Ω) ∩ L_∞ (Ω) to the boundary-value problem for a second-order elliptic system with strong nonlinearity in the case of dimension n≥3. Bibliography: 12 titles. Translated fromProblemy Matematicheskogo Analiza, No. 15, 1995, pp. 47–69.     

The Hadamard Product on Open Sets in the Extended Plane

For two open sets Ω₁, Ω₂ in the extended complex plane, we define a Hadamard product as an operator from H(Ω₁) × H(Ω₂) to H(Ω₁ * Ω₂), where Ω₁ * Ω₂ is the so-called star product. Moreover, we study properties of this product and give applications.     

Controllability in a filled domain for the multidimensional wave equation with a singular boundary control

In accordance with the demands of the so-called local approach to inverse problems, the set of “waves” u^f (·, T) is studied, where u^f (x,t) is the solution of the initial boundary-value problem u_tt−Δu=0 in Ω×(0,T), u|_t<0=0, u|_∂Ω×(0,T)=f, and the (singular) control f runs over the class L²((0,T); H^−m (∂Ω)) (m>0). The following result is established. Let Ω^T={x ∈ Ω : dist(x, ∂Ω)<T)} be a subdomain of Ω ⊂ ℝⁿ (diam Ω<∞) filled with waves by a final instant of time t=T, let T_*=inf{T : Ω^T=Ω} be the time of filling the whole domain Ω. We introduce the notation D_m=Dom((−Δ)^m/2), where (−Δ) is the Laplace operator, Dom(−Δ)=H²(Ω)∩H ₀ ¹ (Ω);D_−m=(D_m)′;D_−m(Ω^T)={y∈D_−m:supp y ⋐ Ω^T. If T<T., then the reachable set R _m ^T ={u^t(·, T): f ∈ L₂((0,T), H^−m (∂Ω))} (∀m>0), which is dense in D_−m(Ω^T), does not contain the class C ₀ ^∞ (Ω^T). Examples of a ∈ C ₀ ^∞ , a ∈ R _m ^T , are presented. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 210, 1994, pp. 7–21. Translated by T. N. Surkova.     

The Besov Space B^{{0,1}}_{1} on Domains

Let Ω be a bounded Lipschitz domain. Define B ^0,1 _{1, r} (Ω) = {f∈L ¹ (Ω): there is an F∈B ^0,1 ₁ (ℝ ⁿ ) such that F|_Ω = f} and B ^0,1 _{1 z} (Ω) = {f∈B ^0,1 ₁ (ℝ ⁿ ) : f = 0 on ℝ ⁿ \}. In this paper, the authors establish the atomic decompositions of these spaces. As by-products, the authors obtained the regularity on these spaces of the solutions to the Dirichlet problem and the Neumann problem of the Laplace equation of ℝ ⁿ ₊. Received June 8, 2000, Accepted October 24, 2000     

The Gleason’s problem for some polyharmonic and hyperbolic harmonic function spaces

Let Ω ⊆ ℝⁿ be a bounded convex domain with C ² boundary. For 0 < p, q ⩽ ∞ and a normal weight φ, the mixed norm space H _k ^p,q,φ (Ω) consists of all polyharmonic functions f of order k for which the mixed norm ∥ · ∥_p,q,φ < ∞. In this paper, we prove that the Gleason’s problem (Ω, a, H _k ^p,q,φ ) is always solvable for any reference point a ∈ Ω. Also, the Gleason’s problem for the polyharmonic φ-Bloch (little φ-Bloch) space is solvable. The parallel results for the hyperbolic harmonic mixed norm space are obtained.     

D p spaces on bounded symmetric domains ofC n

In this paper, the spaceD ^p(Ω) of functions holomorphic on bounded symmetric domain ofC ⁿ is defined. We prove thatH ^p(Ω)⊂D ^p(Ω) if 0<p≤2, andD ^p(Ω)⊂H ^p(Ω) ifp≥2, and both the inclusions are proper. Further, we find that some theorems onH ^p(Ω) can be extended to a wider classD ^p(Ω) for 0<p≤2.     

Torsion in Boundary Coinvariants and K -Theory for Affine Buildings

Let (G, I, N, S) be an affine topological Tits system, and let Γ be a torsion-free cocompact lattice in G. This article studies the coinvariants H ₀(Γ; C(Ω,Z)), where Ω is the Furstenberg boundary of G. It is shown that the class [1] of the identity function in H ₀(Γ; C(Ω, Z)) has finite order, with explicit bounds for the order. A similar statement applies to the K ₀ group of the boundary crossed product C ^*-algebra C(Ω)Γ. If the Tits system has type ? ₂, exact computations are given, both for the crossed product algebra and for the reduced group C ^*-algebra.     

Decompositions of the Hardy space Hz^1（Ω）

We give a decomposition of the Hardy space Hz^1（Ω） into ＂div-curl＂ quantities for Lipschitz domains in R^n. We also prove a decomposition of Hz^1（Ω） into Jacobians det Du, u ∈ W0^1,2 （Ω,R^2） for Ω in R^2. This partially answers a well-known open problem.     

Well-posedness and sharp uniform decay rates at the L 2(Ω)-Level of the Schrödinger equation with nonlinear boundary dissipation

We prove that the Schr?dinger equation defined on a bounded open domain of and subject to a certain attractive, nonlinear, dissipative boundary feedback is (semigroup) well-posed on L₂(Ω) for any n = 1, 2, 3, ..., and, moreover, stable on L₂(Ω) for n = 2, 3, with sharp (optimal) uniform rates of decay. Uniformity is with respect to all initial conditions contained in a given L₂(Ω)-ball. This result generalizes the corresponding linear case which was proved recently in [L-T-Z.2]. Both results critically rely—at the outset—on a far general result of interest in its own right: an energy estimate at the L₂(Ω)-level for a fully general Schr?dinger equation with gradient and potential terms. The latter requires a heavy use of pseudo-differential/micro-local machinery [L-T-Z.2, Section 10], to shift down the more natural H¹(Ω)-level energy estimate to the L₂(Ω)-level. In the present nonlinear boundary dissipation case, the resulting energy estimate is then shown to fit into the general uniform stabilization strategy, first proposed in [La-Ta.1] in the case of wave equations with nonlinear (interior and) boundary dissipation.     

Estimate on the Dimension of Global Attractor for Nonlinear Dissipative Kirchhoff Equation

In this paper,we estimate the dimension of the global attractor for nonlinear dissipative Kirchhoff equation in Hilbert spaces H ₀¹×L ²(Ω) and D(A)×H ₀¹(Ω). Using rescaling technology and linear variation method, we obtain the upper bound for its Hausdorff and fractal dimensions.     

Some results related to interpolation on hardy spaces of regular martingales

Let (Ω,F, P) be a probability space and {F _n}_n≥0 a regular increasing sequence of sub-σ-fields ofF. LetH ₁(Ω) be the usual Hardy space ofF _n-martingales. We show that the couple (H ₁(Ω),L _∞(Ω)) is a partial retract of (L ₁(Ω),L _∞(Ω)). It is also proved that (L _p(Ω),BMO(Ω)) is a partial retract of (L _p(Ω),L _∞(Ω)) for all 1<p<∞.     

Exact controllability for the wave equation with variable coefficients

We consider in this paper the evolution systemy″−Ay=0, whereA =∂ _i(a_ij∂_j) anda _ij ∈C ¹ (ℝ⁺;W ^1,∞ (Ω)) ∩W ^1,∞ (Ω × ℝ⁺), with initial data given by (y ₀,y ₁) ∈L ²(Ω) ×H ⁻¹ (Ω) and the nonhomogeneous conditiony=v on Γ ×]0,T[. Exact controllability means that there exist a timeT>0 and a controlv such thaty(T, v)=y′(T, v)=0. The main result of this paper is to prove that the above system is exactly controllable whenT is “sufficiently large”. Moreover, we obtain sharper estimates onT.     

Regularity of the attractor for 3-D complex Ginzburg-Landau equation

In this paper,the existence of global attractor for 3-D complex Ginzburg Landau equation is considered.By a decomposition of solution operator,it is shown that the global attractor A_i in H~i(Ω) is actually equal to a global attractor Aj in H~j(Ω)(i≠j,i,j = 1,2,…m).     

Analytic Semigroups on H 1 (Ω) Generated by Degenerate Elliptic Operators

One proves that Ay=aΔ y+b⋅ \nabla y , D(A)={y∈ H ¹ (Ω ); aΔ y+b⋅ \nabla y∈ H ₀ ¹ (Ω ), \sqrt a Δ y∈ L ² (Ω )} generates, under suitable conditions on a and b , a C ₀ -analytic semigroup on H ¹ (Ω) . September 5, 1999     

An Obstacle Control Problem with a Source Term

   Abstract. An optimal control problem for an elliptic variational inequality with a source term is considered. The obstacle is the control, and the goal is to keep the solution of the variational inequality close to the desired profile while the H ¹ norm of the obstacle is not too large. The addition of the source term strongly affects the needed compactness result for the existence of a minimizer.     

On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities

We study the Navier-Stokes equations for compressible barotropic fluids in a bounded or unbounded domain Ω of R³. We first prove the local existence of solutions (ρ,u) in C([0,T_*]; (ρ^∞ +H³(Ω)) × under the assumption that the data satisfies a natural compatibility condition. Then deriving the smoothing effect of the velocity u in t>0, we conclude that (ρ,u) is a classical solution in (0,T_**)×Ω for some T_** ∈ (0,T_*]. For these results, the initial density needs not be bounded below away from zero and may vanish in an open subset (vacuum) of Ω.