Strong approximation ofGSBV functions by piecewise smooth functions

Let Ω be an open and bounded subset ofR ⁿ with locally Lipschitz boundary. We prove that the functionsv∈SBV(Ω,R ^m ) whose jump setS _vis essentially closed and polyhedral and which are of classW ^{k, ∞} (S _v,R ^m) for every integerk are strongly dense inGSBV ^p(Ω,R ^m ), in the sense that every functionu inGSBV ^p(Ω,R ^m ) is approximated inL ^p(Ω,R ^m ) by a sequence of functions {v _k{_j∈N with the described regularity such that the approximate gradients ∇v _jconverge inL ^p(Ω,R ^nm ) to the approximate gradient ∇u and the (n−1)-dimensional measure of the jump setsS _{v j} converges to the (n−1)-dimensional measure ofS _u. The structure ofS _v can be further improved in casep≤2.

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Sunto Sia Ω un aperto limitato diR ⁿ con frontiera localmente Lipschitziana. In questo lavoro si dimostra che le funzioniv∈SBV(Ω,R ^m ) con insieme di saltoS _v essenzialmente chiuso e poliedrale che sono di classeW ^{k, ∞} (S _v,R ^m ) per ogni interok sono fortemente dense inGSBV ^p(Ω,R ^m ), nel senso che ogni funzioneu∈GSBV ^p(Ω,R ^m ) è approssimata inL ^p(Ω,R ^m ) da una successione di funzioni {v _j}_j∈N con la regolaritá descritta tali che i gradienti approssimati ∇v _jconvergono inL ^p(Ω,R ^nm ) al gradiente approssimato ∇u e la misura (n−1)-dimensionale degli insiemi di saltoS _{v j}converge alla misura (n−1)-dimensionale diS _u. La struttura diS _vpuó essere migliorata nel caso in cuip≤2.

     

Sharp stability results for almost conformal maps in even dimensions

Let Ω, ⊂R ⁿ and n ≥ 4 be even. We show that if a sequence {u^j} in W^1,n/2(Ω;R ⁿ) is almost conformal in the sense that dist (∇u^j,R ⁺SO(n)) converges strongly to 0 in L^n/2 and if u^j converges weakly to u in W^1,n/2, then u is conformal and ∇u^j → ∇u strongly in L _loc ^q for all 1 < -q < n/2. It is known that this conclusion fails if n/2 is replaced by any smaller exponent p. We also prove the existence of a quasiconvex function f(A) that satisfies 0 ≤ f(A) ≤ C (1 + |A|^n/2) and vanishes exactly onR ⁺ SO(n). The proof of these results involves the Iwaniec-Martin characterization of conformal maps, the weak continuity and biting convergence of Jacobians, and the weak-L¹ estimates for Hodge decompositions.     

Sobolev functions whose inner trace at the boundary is zero

Let Ω⊂R ⁿ be an arbitrary open set. In this paper it is shown that if a Sobolev functionf∈W ^1,p (Ω) possesses a zero trace (in the sense of Lebesgue points) on ϖΩ, thenf is weakly zero on ϖΩ in the sense thatf∈W ₀ ^1,p (Ω).     

LOWER SEMICONTINUITY OF A CLASS OF FUNCTIONALS IN SBVH

In this paper lower semicontinuity of the functional I(u)=∫ _Ω f(x,u,Δ _Hu)dx is investigated for f being a Carathéodory function defined on H ⁿ × R × R²ⁿ and for u∈SBV _H (Ω), where H ⁿ is the Heisenberg group with dimension 2n+1, Ω∩H ⁿ is an open set and ∇ Hu denotes the approximate derivative of the absolute continuous part D ^a _Hu with respect to D _Hu. In addition, a Lusin type approximation theorem for a SBV _H function is proved.     

Global existence and L ∞ estimates of solutions for a quasilinear parabolic system

In this paper, we study the global existence, L^∞ estimates and decay estimates of solutions for the quasilinear parabolic system u_t = div (|∇ u|^m ∇ u) + f(u, v), v_t = div (|∇ v|^m ∇ v) + g(u,v) with zero Dirichlet boundary condition in a bounded domain Ω ⊂ R^N. In particular, we find a critical value for the existence and nonexistence of global solutions to the equation u_t = div (|∇ u|^m ∇ u) + λ |u|^{α - 1} u.     

Cluster sets for sobolev functions and quasiminimizers

In this paper, we study cluster sets and essential cluster sets for Sobolev functions and quasiharmonic functions (i.e., continuous quasiminimizers). We develop their basic theory with a particular emphasis on when they coincide and when they are connected. As a main result, we obtain that if a Sobolev function u on an open set Ω has boundary values f in Sobolev sense and f |_∂Ω is continuous at x ₀ ∈ ∂Ω, then the essential cluster set (u, x ₀,Ω) is connected. We characterize precisely in which metric spaces this result holds. Further, we provide some new boundary regularity results for quasiharmonic functions. Most of the results are new also in the Euclidean case.     

Non-existence of nodal solution form-Laplace equation involving critical Sobolev exponents

In this paper we study the non-existence of nodal solutions for critical Sobolev exponent problem-div(|∇u| ^m−2∇u)=|u| ^p-1 u+|u| ^q-1 u inB(R)u = 0 on ∂B(R) whereB(R) is a ball of radiusR in ℝⁿ.     

Onp-harmonic functions in the complex plane and curvature

In the complex plane thep-harmonic equation div(|∇u| ^p−2∇u) = 0, 1 <p < ∞, exhibits some features reminiscent of Function Theory. Our results about curvature in this structure complement known facts about minimal surfaces and harmonic functions. Quasiregular mappings are used.     

Three-term asymptotics for the boundary layers of semilinear elliptic eigenvalue problems

We consider the nonlinear eigenvalue problem −Δu=λ f(u) in Ω u=0 on ∂Ω, where Ω is a ball or an annulus in R^N (N ≥ 2) and λ > 0 is a parameter. It is known that if λ >> 1, then the corresponding positive solution u_λ develops boundary layers under some conditions on f. We establish the asymptotic formulas for the slope of the boundary layers of u_λ with the exact second term and the ‘optimal’ estimate of the third term.     

On elliptic equations and systems with critical growth in dimension two

We consider nonlinear elliptic equations of the form −Δu = g(u) in Ω, u = 0 on ∂Ω, and Hamiltonian-type systems of the form −Δu = g(v) in Ω, −Δv = f(u) in Ω, u = 0 and v = 0 on ∂Ω, where Ω is a bounded domain in ℝ² and f, g ∈ C(ℝ) are superlinear nonlinearities. In two dimensions the maximal growth (= critical growth) of f and g (such that the problem can be treated variationally) is of exponential type, given by Pohozaev-Trudinger-type inequalities. We discuss existence and nonexistence results related to the critical growth for the equation and the system. A natural framework for such equations and systems is given by Sobolev spaces, which provide in most cases an adequate answer concerning the maximal growth involved. However, we will see that for the system in dimension 2, the Sobolev embeddings are not sufficiently fine to capture the true maximal growths. We will show that working in Lorentz spaces gives better results. Dedicated to Professor S. Nikol’skii on the occasion of his 100th birthday     

Interpolation inequalities and decay estimates for derivatives

Under the only assumption of the cone property for a given domain Ω⊂R ⁿ, it is proved that interpolation inequalities for intermediate derivatives of functions in the Sobolev spaces W^m,p (Ω) or even in some weighted Sobolve spaces W _w ^m,p (Ω) still hold. That is, the usual additional restrictions that Ω is bounded or has the uniform cone property are both removed. The main tools used are polynomial inequalities, by which it is also obtained pointwise version interpolation inequalities for smooth and analytic functions. Such pointwise version inequalities give explicit decay estimates for derivatives at infinity in unbounded domains which have the cone property. As an application of the decay estimates, a previous result on radial basis function approximation of smooth functions is extended to the derivative-simultaneous approximation.     

Everywhere H?lder continuity of the spatial gradient of weak solutions to nonlinear parabolic systems

We consider weak solutions to the parabolic system ∂u ⁱ∂t−D _α A _i ^α (∇u)=B _i(∇u) in (i=1,...,) (Q=Ω×(0,T), R ⁿ a domain), where the functionsB _i may have a quadratic growth. Under the assumptionsn≤2 and ∇u ɛL _loc ^4+δ (Q; R ^nN ) (δ>0) we prove that ∇u is locally H?lder continuous inQ.     

REGULARITY OF POISSON EQUATION IN SOME LOGARITHMIC SPACE

In this note, the regularity of Poisson equation -△u = f with f lying in logarithmic function space Lp(LogL)a(Ω)(1＜p ＜∞, a ∈ R) is studied. The result of the note generalizes the W2,p estimate of Poisson equation in Lp(Ω).     

Overdetermined problems with possibly degenerate ellipticity, a geometric approach

Given an open bounded connected subset Ω of ℝⁿ, we consider the overdetermined boundary value problem obtained by adding both zero Dirichlet and constant Neumann boundary data to the elliptic equation −div(A(|∇u|)∇u)=1 in Ω. We prove that, if this problem admits a solution in a suitable weak sense, then Ω is a ball. This is obtained under fairly general assumptions on Ω and A. In particular, A may be degenerate and no growth condition is required. Our method of proof is quite simple. It relies on a maximum principle for a suitable P-function, combined with some geometric arguments involving the mean curvature of ∂Ω.     

Extreme Jensen measures

Let Ω be an open subset of R ^d , d≥2, and let x∈Ω. A Jensen measure for x on Ω is a Borel probability measure μ, supported on a compact subset of Ω, such that ∫u  dμ≤u(x) for every superharmonic function u on Ω. Denote by J _x (Ω) the family of Jensen measures for x on Ω. We present two characterizations of ext(J _x (Ω)), the set of extreme elements of J _x (Ω). The first is in terms of finely harmonic measures, and the second as limits of harmonic measures on decreasing sequences of domains. This allows us to relax the local boundedness condition in a previous result of B. Cole and T. Ransford, Jensen measures and harmonic measures, J. Reine Angew. Math. 541 (2001), 29–53. As an application, we give an improvement of a result by Khabibullin on the question of whether, given a complex sequence {α _n } _n=1 ^∞ and a continuous function , there exists an entire function f≢0 satisfying f(α _n )=0 for all n, and |f(z)|≤M(z) for all z∈C.     

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     

A Fully Nonlinear Nonhomogeneous Neumann Problem

Let Ω be an open bounded set in ℝ^N, N≥3, with connected Lipschitz boundary ∂Ω and let a(x,ξ) be an operator of Leray–Lions type (a(⋅,∇u) is of the same type as the operator |∇u|^p−2∇u, 1<p<N). If τ is the trace operator on ∂Ω, [φ] the jump across ∂Ω of a function φ defined on both sides of ∂Ω, the normal derivative ∂/∂ν_a related to the operator a is defined in some sense as 〈a(⋅,∇u),ν〉, the inner product in ℝ^N, of the trace of a(⋅,∇u) on ∂Ω with the outward normal vector field ν on ∂Ω. If β and γ are two nondecreasing continuous real functions everywhere defined in ℝ, with β(0)=γ(0)=0, f∈L¹(ℝ^N), g∈L¹(∂Ω), we prove the existence and the uniqueness of an entropy solution u for the following problem,

The Discrete and Classical Dirichlet Problem

Let W ì \mathbbR^d{\Omega \subset \mathbb{R}^d} be some bounded domain with reasonable boundary and let f be a continuous function on the complement Ω ^c . We can construct an unique continuous function u that is harmonique on Ω and u = f on Ω ^c . Similarly, u _d is the unique function on the lattice points such that for each lattice point of Ω satisfies the “average” property with respect to its nearest neighbours and u _d = f on Ω ^c . In this paper when ∂Ω is Lipschitz I give a “best possible” estimate of ||u − u _d ||_∞.     

Solutions of elliptic equations involving critical Sobolev exponents with neumann boundary conditions

We consider the problem −Δu=|u| ^p−1u+λu in Ω with on δΩ, where Ω is a bounded domain inR ^N ,p=(N+2)/(N−2) is the critical Sobolev exponent,n the outward pointing normal and λ a constant. Our main result is that if Ω is a ball inR ^N , then for every λ∈R the problem admits infinitely many solutions. Next we prove that for every bounded domain Ω inR ³, symmetric with respect to a plane, there exists a constant μ>0 such that for every λ<μ this problem has at least one non-trivial solution. This work was supported by the Paris VI-Leiden exchange program Supported by the Netherlands organisation for scientific research NWO, under number 611-306-016.     

On convergence of gradient-dependent integrands

We study convergence properties of {υ(∇u _k )}k∈ℕ if υ ∈ C(ℝ ^m×m ), |υ(s)| ⩽ C(1+|s| ^p ), 1 < p < + ∞, has a finite quasiconvex envelope, u _k → u weakly in W ^1,p (Ω; ℝ ^m ) and for some g ∈ C(Ω) it holds that ∫_Ω g(x)υ(∇u _k (x))dx → ∫_Ω g(x)Qυ(∇u(x))dx as k → ∞. In particular, we give necessary and sufficient conditions for L ¹-weak convergence of {det ∇u _k } _k∈ℕ to det ∇u if m = n = p. Dedicated to Jiří V. Outrata on the occasion of his 60th birthday This work was supported by the grants IAA 1075402 (GA AV ČR) and VZ6840770021 (MŠMT ČR).