Regularity for solutions to anisotropic obstacle problems

For Ω a bounded subset of R n,n 2,ψ any function in Ω with values in R∪{±∞}andθ∈W1,(q i)(Ω),let K(q i)ψ,θ(Ω)={v∈W1,(q i)(Ω):vψ,a.e.and v-θ∈W1,(q i)0(Ω}.This paper deals with solutions to K(q i)ψ,θ-obstacle problems for the A-harmonic equation-divA(x,u(x),u(x))=-divf(x)as well as the integral functional I(u;Ω)=Ωf(x,u(x),u(x))dx.Local regularity and local boundedness results are obtained under some coercive and controllable growth conditions on the operator A and some growth conditions on the integrand f.     

Regularity of weak solutions of one-dimensional two-phase Stefan problems

Sunto Si considera il problema di Stefan unidimensionale a due fasi e si dimostra l'esistenza di soluzioni classiche sotto ipotesi minimali sui dati (continuità a tratti e limitatezza). Nelle stesse ipotesi si dimostra che tali soluzioni dipendono in modo continuo dai dati, conseguendo un risultato che è più generale anche di quello noto per le soluzioni deboli.

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Entrata in Redazione il 22 dicembre 1976.

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Work partially supported by the Italian C.N.R.     

Regularity of variational solutions to linear boundary value problems in Lipschitz domains

In a bounded Lipschitz domain in ?ⁿ, we consider a second-order strongly elliptic system with symmetric principal part written in divergent form. We study the Neumann boundary value problem in a generalized variational (or weak) setting using the Lebesgue spaces H _p ^σ (Ω) for solutions, where p can differ from 2 and σ can differ from 1. Using the tools of interpolation theory, we generalize the known theorem on the regularity of solutions, in which p = 2 and {σ ? 1} < 1/2, and the corresponding theorem on the unique solvability of the problem (Savaré, 1998) to p close to 2. We compare this approach with the nonvariational approach accepted in numerous papers of the modern theory of boundary value problems in Lipschitz domains. We discuss the regularity of eigenfunctions of the Dirichlet, Neumann, and Poincaré-Steklov spectral problems.     

Regularity of semi-stable solutions to fourth order nonlinear eigenvalue problems on general domains

We examine the fourth order problem $\Delta ^2 u = \lambda f(u) $ in $ \Omega $ with $ \Delta u = u =0 $ on $ {\partial \Omega }$ , where $ \lambda > 0$ is a parameter, $ \Omega $ is a bounded domain in $\mathbb{R }^N$ and where $f$ is one of the following nonlinearities: $ f(u)=e^u$ , $ f(u)=(1+u)^p $ or $ f(u)= \frac{1}{(1-u)^p}$ where $ p>1$ . We show the extremal solution is smooth, provided $$\begin{aligned} N < 2 + 4 \sqrt{2} + 4 \sqrt{ 2 - \sqrt{2}} \approx 10.718 \text{ when} f(u)=e^u, \end{aligned}$$ and $$\begin{aligned} N < \frac{4p}{p-1} + \frac{4(p+1)}{p-1} \left( \sqrt{ \frac{2p}{p+1}} + \sqrt{ \frac{2p}{p+1} - \sqrt{ \frac{2p}{p+1}}} - \frac{1}{2} \right) \end{aligned}$$ when $ f(u)=(u+1)^p$ . New results are also obtained in the case where $ f(u)=(1-u)^{-p}$ . These are substantial improvements to various results on critical dimensions obtained recently by various authors. To do that, we derive a new stability inequality satisfied by minimal solutions of the above equation, which is more amenable to estimates as it allows a method of proof reminiscent of the second order case.     

Regularity up to the boundary of solutions to boundary-value problems of the theory of generalized Newtonian liquids

Boundary-value problems describing the stationary flow of a generalized Newtonian liquid are considered. The regularity of solutions to such problems is studied near the boundary. The W ₂ ² -estimate for a solution and the partial regularity of the strain velocity tensor are established. In the two-dimensional case, the complete regularity of the strain velocity tensor is also proved. Bibliography: 12 titles. Translated fromProblemy Matematicheskogo Analiza, No. 16. 1997, pp. 239–265.     

State constrained control problems for parabolic systems: Regularity of optimal solutions

Quadratic control problems for parabolic equations withstate constraints are considered. Regularity (smoothness) of the optimal solution is investigated. It is shown that the optimal control is continuous in time with the values inL ₂() and its time derivative belongs toL ₂[OT×].Research partially supported by National Aeronautics and Space Administration under Grant No. NSG 4015.     

Fractional powers of operators corresponding to coercive problems in Lipschitz domains

Let Ω be a bounded Lipschitz domain in ? ⁿ , n ? 2, and let L be a second-order matrix strongly elliptic operator in Ω written in divergence form. There is a vast literature dealing with the study of domains of fractional powers of operators corresponding to various problems (beginning with the Dirichlet and Neumann problems) with homogeneous boundary conditions for the equation Lu = f, including the solution of the Kato square root problem, which arose in 1961. Mixed problems and a class of problems for higher-order systems have been covered as well. We suggest a new abstract approach to the topic, which permits one to obtain the results that we deem to be most important in a much simpler and unified way and cover new operators, namely, classical boundary operators on the Lipschitz boundary Γ = ?Ω or part of it. To this end, we simultaneously consider two well-known operators associated with the boundary value problem.     

Entire solutions of completely coercive quasilinear elliptic equations

A famous theorem of Sergei Bernstein says that every entire solution u=u(x), x∈R², of the minimal surface equation

     

Regularity of solutions to the perturbed conservation laws

A kind of regularity for the mild solution of perturbed conservation laws is proposed. This regularity is described in term of variations measured in the L¹-norm. A dissipativity condition from the semigroup approach is used to show that the mild solution stays within a class of bounded variation in this sense of regularity. This shows that this class of functions is an invariant of the semigroup. The same analysis carries over to the periodic problem. The class of boundedL¹-variation functions used here can be normed to give a Banach space structure. It also has an analogue with the space of Lipschitz functions     

Regularity of the Koch-Tataru solutions to Navier-Stokes system

In this paper,we shall prove that the Koch-Tataru solution u to the incompressible Navier-Stokes equations in Rd satisfies the decay estimates involving some borderline Besov norms with d 3.Moreover,u has a unique trajectory which is Hlder continuous with respect to the space variables.     

Minima of some non convex non coercive problems

We give here an existence result of minimizers for a class of one dimensional integrals of the Calculus of Variations with non convex, non coercive integrands.This work was performed as part of a national research project supported by M.U.R.S.T. (40%).Authors are members of G.N.A.F.A., C.N.R.     

Optimal stopping problems with discontinous reward: Regularity of the value function and viscosity solutions

In this paper we show that the solution of a stochastic boundary value problem with additive noise and with a completely nonlinear drift is a Markov field if only if the boundary condition is an initial or a final type condition