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.     

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.

     

Some remarks on the identity between a variational integral and its relaxed functional

Summary Letf: (x, z)∈R ⁿ×Rⁿ→f(x, z)∈[0, +∞] be measurable inx and convex inz. It is proved, by an example, that even iff verifies a condition as|z| ^p≤f(x, z)≤Λ(a(x)+|z|^q) with 1<p<q,a∈L _loc ^s (R ⁿ),s>1, the functional that isL ¹(Ω)-lower semicontinuous onW ^1,1(Ω), does not agree onW ^1,1(Ω) with its relaxed functional in the topologyL ¹(Ω) given by inf

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Riassunto Siaf: (x, z)∈R ⁿ×Rⁿ→f(x, z)∈[0, +∞] misurabile inx e convessa inz. Si mostra con un esempio che anche sef verifica una condizione del tipo|z| ^p≤f(x, z)≤Λ(a(x)+|z|^q) con 1<p<q,a∈L _loc ^s (R ⁿ),s>1, il funzionale , che èL ¹(Ω)-semicontinuo inferiormente suW ^1,1(Ω), non coincide suW ^1,1(Ω) con il suo funzionale rilassato nella topologiaL ¹(Ω) definito da inf

     

On minimizing partitions with infinitely many components

It is known [8] that, wheng ∈L ⁿ (Ω) (Ω open and bounded inR ⁿ , with ≪regular≫ boundary∂Ω), any minimizer (K, w) of the functional among relatively closed subsetsC ofΩ and piecewise-constant functionsu onΩ/C, gives rise to a finite decomposition ofΩ/K. Here we exhibit a piecewise-constant functiong on the unit diskD ofR ², with radial symmetry, such thatg ∈L ^q (D) for all 1 ⩽q < 2 and the unique minimizer of F has infinitely many components. We also fill a gap occurred in the proof of Proposition 5.2 of [8].

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Sunto è noto [8] che quandog ∈L ⁿ (Ω (Ω aperto limitato diR ⁿ , con frontiera sufficientemente regolare) i minimi (K, w) del funzionale , doveC è relativamente chiuso in Ω eu è costante a tratti suΩ/C, danno luogo a decomposizioni finite diΩ/K. In questo lavoro mostriamo un controesempio relativo ad un datog ∈L ^q (D) per ogni 1 ⩽q < 2 (D è il disco unitario diR ²), a simmetria radiale e costante a tratti. Viene inoltre corretto un errore occorso nella dimostrazione della Prop. 5.2 di [8].

     

Pipes, Cigars, and Kreplach: the Union of Minkowski Sums in Three Dimensions

Let Ω be a set of pairwise-disjoint polyhedral obstacles in R ³ with a total of n vertices, and let B be a ball in R ³. We show that the combinatorial complexity of the free configuration space F of B amid Ω:, i.e., (the closure of) the set of all placements of B at which B does not intersect any obstacle, is O(n ^2+ε ), for any ε >0; the constant of proportionality depends on ε. This upper bound almost matches the known quadratic lower bound on the maximum possible complexity of F . The special case in which Ω is a set of lines is studied separately. We also present a few extensions of this result, including a randomized algorithm for computing the boundary of F whose expected running time is O(n ^2+ε ). Received July 6, 1999, and in revised form April 25, 2000. Online publication August 18, 2000.     

Punishing Factors for Finitely Connected Domains

Let Ω and Π be two finitely connected hyperbolic domains in the complex plane \Bbb C{\Bbb C} and let R(z, Ω) denote the hyperbolic radius of Ω at z and R(w, Π) the hyperbolic radius of Π at w. We consider functions f that are analytic in Ω and such that all values f(z) lie in the domain Π. This set of analytic functions is denoted by A(Ω, Π). We prove among other things that the quantities C_n(W,P) := sup_{f ? A(W,P)}sup_{z ? W}\frac|f⁽ⁿ⁾(z)| R(f(z),P)n! (R(z,W))ⁿC_n(\Omega,\Pi)\,:=\,\sup_{f\in A(\Omega,\Pi)}\sup_{z\in \Omega}\frac{\vert f^{(n)}(z)\vert\,R(f(z),\Pi)}{n!\,(R(z,\Omega))^n} are finite for all n ? \Bbb N{n \in {\Bbb N}} if and only if ∂Ω and ∂Π do not contain isolated points.     

Harmonic maps from manifolds of <Emphasis Type="Italic">L</Emphasis><Superscript>∞</Superscript>-Riemannian metrics

For a bounded domain Ω ⊂ R ⁿ endowed with L ^∞-metric g, and a C ⁵-Riemannian manifold (N, h) ⊂ R ^k without boundary, let u ∈ W ^1,2(Ω, N) be a weakly harmonic map, we prove that (1) u ∈ C ^α (Ω, N) for n = 2, and (2) for n ≥ 3, if, in additions, g ∈ VMO(Ω) and u satisfies the quasi-monotonicity inequality (1.5), then there exists a closed set Σ ⊂ Ω, with H ^n-2(Σ) = 0, such that for some α ∈ (0, 1). C. Y. Wang Partially supported by NSF.     

Nearly flat almost monotone measures are big pieces of lipschitz graphs

A concentrated (ξ, m) almost monotone measure inR ⁿ is a Radon measure Φ satisfying the two following conditions: (1) Θ ^m (Φ,x)≥1 for every x ∈spt (Φ) and (2) for everyx ∈R ⁿ the ratioexp [ξ(r)]r^−mΦ(B(x,r)) is increasing as a function of r>0. Here ξ is an increasing function such thatlim _r→0-ξ(r)=0. We prove that there is a relatively open dense setReg (Φ) ∋spt (Φ) such that at each x∈Reg(Φ) the support of Φ has the following regularity property: given ε>0 and λ>0 there is an m dimensional spaceW ⊂R ⁿ and a λ-Lipschitz function f from x+W into x+W^‖ so that (100-ε)% ofspt(Φ) ∩B (x, r) coincides with the graph of f, at some scale r>0 depending on x, ε, and λ.     

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.     

Indices of Toeplitz tuples on pseudoregular domains

This paper proves an index theorem of Toeplitz tuples on pseudoregular domains in Cn. Geometrically, the index of Toeplitz tuple TΦn is (-1)n time wrapping number of Φn around the origin. As one of the applications of the index theorem, we completely characterize the automorphism groups of Toeplitz algebras on Poincaré domain. As another application, it is shown that C*(Ω)C*(Bn) for every Poincare domain Ω in Cn(n≠2). It is also noticed that C*(Ω)C*(B2) if and only if the Poincaré conjecture is true for Ω.     

Lower bounds at infinity of solutions of partial differential equations in the exterior of a proper cone

An extension of a classical theorem of Rellich to the exterior of a closed proper convex cone is proved: Let Γ be a closed convex proper cone inR ⁿ and −Γ′ be the antipodes of the dual cone of Γ. Let be a partial differential operator with constant coefficients inR ⁿ, whereQ(ζ)≠0 onR ⁿ−iΓ′ andP _i is an irreducible polynomial with real coefficients. Assume that the closure of each connected component of the set {ζ∈R ⁿ−iΓ′;P _j(ζ)=0, gradP _j(ζ)≠0} contains some real point on which gradP _j≠0 and gradP _j∉Γ∪(−Γ). LetC be an open cone inR ⁿ−Γ containing both normal directions at some such point, and intersecting each normal plane of every manifold contained in {ξ∈R ⁿ;P(ξ)=0}. Ifu∈ℒ′∩L _loc ² (R ⁿ−Γ) and the support ofP(−i∂/∂x)u is contained in Γ, then the condition implies that the support ofu is contained in Γ.     

Potential space estimates in local Hardy spaces for Green potentials in convex domains

Let Ω be a bounded co.nvex domain in Rn(n≥3) and G(x,y) be the Green function of the Laplace operator -△ on Ω. Let hrp(Ω) = {f ∈ D'(Ω) :(E)F∈hp(Rn), s.t. F|Ω = f}, by the atom characterization of Local Hardy spaces in a bounded Lipschitz domain, the bound of f→(△)2(Gf) for every f ∈ hrp(Ω) is obtained, where n/(n 1)＜p≤1.     

Uniform harmonic approximation with continuous extension to the boundary

Let Ω be an open set in ℝ ⁿ andE be a relatively closed subset of Ω. Further, letC _e(E) be the collection of real-valued continuous functions onE which extend continuously to the closure ofE in ℝ ⁿ . We characterize those pairs (Ω,E) which have the following property: every function inC _e(E) which is harmonic onE ⁰ can be uniformly approximated onE by functions which are harmonic on Ω and whose restrictions toE belong toC _e(E).     

Existence and multiplicity of solutions for a nonlinear Neumann problem

We consider a Neumann problem of the type -εΔu+F ^′(u(x))=0 in an open bounded subset Ω of R ⁿ , where F is a real function which has exactly k maximum points. Using Morse theory we find that, for ε suitably small, there are at least 2k nontrivial solutions of the problem and we give some qualitative information about them. Received: October 30, 1999 Published online: December 19, 2001     

Local mappings on spaces of differentiable functions

We study conditions under which a functionalF(u, B) defined for everyu∈C ^k(Ω;R ^m ) and every Borel subsetB of Ω admits the integral representation

Sufficient conditions on the existence of trapped modes in problems of the linear theory of surface waves

An asymptotic model is found for the Neumann problem for the second-order differential equation with piecewise constant coefficients in a composite domain Ω∪ω, which are small, of order ε, in the subdomain ω. Namely, a domain Ω(ε) with a singular perturbed boundary is constructed, the solution for which provides a two-term asymptotic, that is, of increased accuracy O(ε²| log ε|^3/2), approximation to the restriction to Ω of the solution of the original problem. As opposed to other singularly perturbed problems, in the case of contrasting stiffness, the modeling requires the construction of a contour ∂Ω(ε) with ledges, i.e., with boundary fragments of curvature O(ε⁻¹). Bibliography: 33 titles.     

A boundary-value problem for the biharmonic equation and the iterated Laplacian in a 3D-domain with an edge

Let Ω be a domain with piecewise smooth boundary. In general, it is impossible to obtain a generalized solution u ∈ W ₂ ² (Ω) of the equation Δ _x ² u = f with the boundary conditions u = Δ_xu = 0 by solving iteratively a system of two Poisson equations under homogeneous Dirichlet conditions. Such a system is obtained by setting v = −Δu. In the two-dimensional case, this fact is known as the Sapongyan paradox in the theory of simply supported polygonal plates. In the present paper, the three-dimensional problem is investigated for a domain with a smooth edge Γ. If the variable opening angle α ∈ C^∞(Γ) is less than π everywhere on the edge, then the boundary-value problem for the biharmonic equation is equivalent to the iterated Dirichlet problem, and its solution u inherits the positivity preserving property from these problems. In the case α ∈ (π 2π), the procedure of solving the two Dirichlet problems must be modified by permitting infinite-dimensional kernel and co-kernel of the operators and determining the solution u ∈ W ₂ ² (Ω) by inverting a certain integral operator on the contour Γ. If α(s) ∈ (3π/2,2π) for a point s ∈ Γ, then there exists a nonnegative function f ∈ L₂(Ω) for which the solution u changes sign inside the domain Ω. In the case of crack (α = 2π everywhere on Γ), one needs to introduce a special scale of weighted function spaces. In this case, the positivity preserving property fails. In some geometrical situations, the problems on well-posedness for the boundary-value problem for the biharmonic equation and the positivity property remain open. Bibliography: 46 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 336, 2006, pp. 153–198.     

On semi-fredholm properties of a boundary value problem inR + n

The paper considers a boundary value problem with the help of the smallest closed extensionL ^∼ :H _k →H _{k 0}×B _{h 1}×...×B _{h N} of a linear operatorL :C ₍₀₎ ^∞ (R ₊ ⁿ ) →L(R ₊ ⁿ )×L(R ⁿ⁻¹)×...×L(R ⁿ⁻¹). Here the spacesH _k (the spaces ℬ _h ) are appropriate subspaces ofD′(R ₊ ⁿ ) (ofD′(R ⁿ⁻¹), resp.),L(R ₊ ⁿ ) andC ₍₀₎ ^∞ (R ₊ ⁿ )) denotes the linear space of smooth functionsR ⁿ →C, which are restrictions onR ₊ ⁿ of a function from the Schwartz classL (fromC ₀ ^∞ , resp.),L(R ⁿ⁻¹) is the Schwartz class of functionsR ⁿ⁻¹ →C andL is constructed by pseudo-differential operators. Criteria for the closedness of the rangeR(L ^∼) and for the uniqueness of solutionsL ^∼ U=F are expressed. In addition, ana priori estimate for the corresponding boundary value problem is established.     

Lip Automorphism Germ and Lip Automorphism

Let X be a metric space, ε^n(X) be the standard trivial Lip n-bundle over X, and Φ be a Lip automorphism germ of ε^n(X). This paper proves that there is a Lip automorphism Φ‘ of ε^n(X) such that the germ of Φ‘ is Φ.     

Non-wandering sets of the powers of maps of a tree

Let T be a tree and let Ω ( f ) be the set of non-wandering points of a continuous map f: T→ T. We prove that for a continuous map f: T→ T of a tree T: ( i) if x∈ Ω( f) has an infinite orbit, then x∈ Ω( fⁿ) for each n∈ ℕ; (ii) if the topological entropy of f is zero, then Ω( f) = Ω( fⁿ) for each n∈ ℕ. Furthermore, for each k∈ ℕ we characterize those natural numbers n with the property that Ω(f^k) = Ω(f^kn) for each continuous map f of T.