On the solvability of the jump problem in clifford analysis

Let Ω be a bounded open and oriented connected subset of ? ⁿ which has a compact topological boundary Γ, let C be the Dirac operator in ? ⁿ , and let ?_0,n be the Clifford algebra constructed over the quadratic space ? ⁿ . An ?_0,n -valued smooth function f : Ω → ?_0,n in Ω is called monogenic in Ω if Df = 0 in Ω. The aim of this paper is to present the most general condition on Γ obtained so far for which a Hölder continuous function f can be decomposed as F ⁺ ? F ^? = f on Γ, where the components F ^± are extendable to monogenic functions in Ω^± with Ω⁺ := Ω, and Ω^? := ? ⁿ \ (Ω ? Γ), respectively.     

An extremal case of the equation of prescribed Weingarten curvature

In this paper we investigate the existence and regularity of solutions to a Dirichlet problem for a Hessian quotient equation on the sphere. The equation in question arises as the determining equation for the support function of a convex surface which is required to meet a given enclosing cylinder tangentially and whose k-th Weingarten curvature is a given function. This is a generalization of a Gaussian curvature problem treated in [13]. Essentially given ${\Omega \subset \mathbb{R}^n}$ we seek a convex function u such that graph(u) has a prescribed k-th curvature ψ and |Du(x)| → ∞ as x → ?Ω. Under certain regularity assumptions on ψ and Ω we are able to demonstrate the existence of a solution whose graph is C ^3,α provided that ${\psi^{-\frac{1}{k}} = \psi^{-\frac{1}{k}}(x, \nu)}$ is convex in x and a certain compatibility condition between ψ| _?Ω and Ω is satisfied.     

Inverse Problems of Boundary Value Problems for Annular Vibrating Membranes with Piecewise Smooth Positive Functions in the Boundary Conditions

The asymptotic expansions of the trace of the heat kernel for small positive t, where λ_ν are the eigenvalues of the negative Laplacian in Rⁿ (n=2 or 3), are studied for a general annular bounded domain Ω with a smooth inner boundary ?Ω₁ and a smooth outer boundary ?Ω₂ where a finite number of piecewise smooth Dirichlet, Neumann and Robin boundary conditions on the components Γ _j (j=1,…,m) of ?Ω₁ and on the components of ?Ω₂ are considered such that and and where the coefficients in the Robin boundary conditions are piecewise smooth positive functions. Some applications of Θ (t) for an ideal gas enclosed in the general annular bounded domain Ω are given.     

Polar factorization and monotone rearrangement of vector-valued functions

Given a probability space (X, μ) and a bounded domain Ω in ?^d equipped with the Lebesgue measure |·| (normalized so that |Ω| = 1), it is shown (under additional technical assumptions on X and Ω) that for every vector-valued function u ∈ L^p (X, μ; ?^d) there is a unique “polar factorization” u = ?Ψs, where Ψ is a convex function defined on Ω and s is a measure-preserving mapping from (X, μ) into (Ω, |·|), provided that u is nondegenerate, in the sense that μ(u^?1(E)) = 0 for each Lebesgue negligible subset E of ?^d. Through this result, the concepts of polar factorization of real matrices, Helmholtz decomposition of vector fields, and nondecreasing rearrangements of real-valued functions are unified. The Monge-Ampère equation is involved in the polar factorization and the proof relies on the study of an appropriate “Monge-Kantorovich” problem.     

A lower bound on the Kobayashi metric near a point of finite type in ℂ n

Let Ω be a bounded domain in ? ⁿ andbΩ smooth pseudoconvex near z₀ ∈bΩ of finite type. Then there are constantsc>0 and ε′>0 such that the Kobayashi metric,K _Ω(z; X), satisfiesK _Ω(z; X)≥c|X|δ(z)^?t′ for allX ∈T _z ^1,0 ? ⁿ in a neighborhood ofz ₀. Here δ(z) denotes the distance fromz tobΩ. As an application, we prove the Hölder continuity of proper holomorphic maps onto pseudoconvex domains.     

Obstacle Problems with Linear Growth: Hölder Regularity for the Dual Solution

For a strictly convex integrand f : ℝⁿ → ℝ with linear growth we discuss the variational problem among mappings u : ℝⁿ ⊃ Ω → ℝ of Sobolev class W¹₁ with zero trace satisfying in addition u ≥ ψ for a given function ψ such that ψ|_∂Ω < 0. We introduce a natural dual problem which admits a unique maximizer σ. In further sections the smoothness of σ is investigated using a special J-minimizing sequence with limit u* ∈ C^1,α (Ω) for which the duality relation holds.     

Probing for electrical inclusions with complex spherical waves

Let a physical body Ω in ?² or ?³ be given. Assume that the electric conductivity distribution inside Ω consists of conductive inclusions in a known smooth background. Further, assume that a subset Γ ? ?Ω is available for boundary measurements. It is proved using hyperbolic geometry that certain information about the location of the inclusions can be exactly recovered from static electric measurements on Γ. More precisely: given a ball B with center outside the convex hull of Ω and satisfying (B? ∩ ?Ω) ? Γ, boundary measurements on Γ with explicitly given Dirichlet data are enough to determine whether B intersects the inclusion. An approximate detection algorithm is introduced based on the theory. Numerical experiments in dimension two with simulated noisy data suggest that the algorithm finds the inclusion‐free domain near Γ and is robust against measurement noise. © 2007 Wiley Periodicals, Inc.     

Best Constants in the Miranda-Agmon Inequalities for Solutions of Elliptic Systems and the Classical Maximum Modulus Principle for Fluid and Elastic Half-spaces

The Dirichlet problem for elliptic systems of the second order with constant real and complex coefficients in the half-space  ^k ₊ = {x = (x ₁,…,x_k ): x_k > 0} is considered. It is assumed that the boundary values of a solution u = (u ₁,…,u _m) have the form ψ ₁ ξ ₁ + · · · + ψ _n ξ _n, 1 ≤ n ≤ m, where ξ ₁,· · ·,ξ _n is an orthogonal system of m-component normed vectors and ψ ₁,· · ·,ψ _n are continuous and bounded functions on ? ^k ₊. We study the mappings [C(? ^k ₊)] ⁿ ? (ψ ₁,…,ψ _n) → u(x) ?  ^m and [C(? ^k ₊)] ⁿ ? (ψ ₁,…,ψ _n) → u(x) ?  ^m generated by real and complex vector valued double layer potentials. We obtain representations for the sharp constants in inequalities between |u(x)| or |(z, u(x))| and ∥u| _xk =₀∥, where z is a fixed unit m-component vector, | · | is the length of a vector in a finite-dimensional unitary space or in Euclidean space, and (·,·) is the inner product in the same space. Explicit representations of these sharp constants for the Stokes and Lamé systems are given. We show, in particular, that if the velocity vector (the elastic displacement vector) is parallel to a constant vector at the boundary of a half-space and if the modulus of the boundary data does not exceed 1, then the velocity vector (the elastic displacement vector) is majorised by 1 at an arbitrary point of the half-space. An analogous classical maximum modulus principle is obtained for two components of the stress tensor of the planar deformed state as well as for the gradient of a biharmonic function in a half-plane.     

Homogenization of the variational inequality corresponding to a problem with rapidly varying boundary conditions

In the present paper, we investigate the asymptotic behavior of the solution of a variational inequality with one-sided constraints on ?-periodically located subsets G _ε belonging to the boundary ?Ω of the domain Ω ? ?³. We construct a limit (homogenized) problem and prove the strong (in H ₁(Ω)) convergence of the solutions of the original inequality to the solution the limit nonlinear boundary-value problem as ? → 0 in the so-called critical case.     

Schauder Estimates for the Single Layer Potential in Hydrodynamics

Let Ω be an open set in ?^N(N ? 3), with compact boundary ?Ω of type C^1,α(?(0,1)). We show that the single layer potential Ef, related to the stationary Stokes system on Ω, belongs to C^1,α(?Ω)^N, provided the source density f belongs to C^α(?Ω)^N. In addition, we prove a related estimate of the function E(f)_?Ω and its tangential derivatives.     

Heat Flow and Perimeter in \boldsymbol{{\mathbb{R}}^m}

Let Ω be an open set in Euclidean space ? ^m with finite perimeter ${\mathcal{P}}(\Omega),$ and with m-dimensional Lebesgue measure |Ω|. It was shown by M. Preunkert that if T(t) is the heat semigroup on L ²(? ^m ) then $H_{\Omega}(t):=\int_{\Omega}T(t)\textbf{1}_{\Omega}(x)dx=|\Omega|-\pi^{-1/2}{\mathcal{P}}(\Omega)t^{1/2}+o(t^{1/2}), \ t\downarrow 0$ . H _Ω(t) represents the amount of heat in Ω if Ω is at initial temperature 1 and if ? ^m ???Ω is at initial temperature 0. In this paper we will compare the quantitative behaviour of H _Ω(t) with the usual heat content Q _Ω(t) associated to the Dirichlet heat semigroup on Ω. We analyse the heat content for horn-shaped open sets of the form Ω(α, Σ)?=?{(x, x′)?∈?? ^m : x′?∈?(1?+?x)^???α Σ, x?>?0}, where α?>?0, and where Σ is an open set in ? ^m???1 with finite perimeter in ? ^m???1, which is star-shaped with respect to 0. For m?≥?3 we find that there are four regimes with very different behaviour depending on α, and a further two limiting cases where logarithmic corrections appear.     

On the instability of resonances in parallel-plane waveguides under local perturbations of the boundary

We study the large-time asymptotics for solutions u( x , t) of the wave equation with Dirichlet boundary data, generated by a time-harmonic force distribution of frequency ω, in a class of domains with non-compact boundaries and show that the results obtained in [11] for a special class of local perturbations of Ω₀ ? ?² × (0,1) can be extended to arbitrary smooth local perturbations Ω of Ω₀. In particular, we prove that u is bounded as t → ∞ if Ω does not allow admissible standing waves of frequency ω in the sense of [8]. This implies in connection with [8]. Theorem 3.1 that the logarithmic resonances of the unperturbed domain Ω₀ at the frequencies ω = πk (k = 1, 2,…) observed in [14] can be simultaneously removed by small perturbations of the boundary. As a main step of our analysis, the determination of admissible solutions of the boundary value problem ΔU + κ²U = ? f in Ω, U = 0 on ?Ω is reduced to a compact operator equation.     

Existence results for quasilinear elliptic equations with multivalued nonlinear terms

In this paper we study the existence of solutions u ∈ \({{W}^{1,p}_{0}}\) (Ω) with △ _p u ∈ L ²(Ω) for the Dirichlet problem 1 $$ \left\{ \begin{array} [c]{l}-\triangle_{p}u\left( x\right) \in-\partial{\Phi}\left( u\left( x\right) \right) +G\left( x,u\left( x\right) \right) ,x\in{\Omega},\\ u\mid_{\partial{\Omega}}=0, \end{array} \right. $$ where Ω ? R ^N is a bounded open set with boundary ?Ω, △ _p stands for the p?Laplace differential operator, ?Φ denotes the subdifferential (in the sense of convex analysis) of a proper convex and lower semicontinuous function Φ and G : Ω × R → 2^R is a multivalued map. We prove two existence results: the first one deals with the case where the multivalued map u ? G(x, u) is upper semicontinuous with closed convex values and the second one deals with the case when u ? G(x, u) is lower semicontinuous with closed (not necessarily convex) values.     

Regularity of hyperbolic equations underL 2(0,T; L 2 (Γ))-Dirichlet boundary terms

With Ω an open bounded domain inR ⁿ with boundary Γ, letf(t; f ₀,f ₁;u) be the solution to a second order linear hyperbolic equation defined on Ω, under the action of the forcing termu(t) applied in the Dirichlet B.C., and with initial dataf ₀ ∈L ₂ (Ω) andf ₁ ∈H ^?1 (Ω). In a previous paper [6], we proved (among other things) that the mapu → f ? f _t , from the Dirichlet input into the solution is continuous fromL ₂(0,T; L ₂ (Γ)) intoL ₂(0,T; L ₂(Ω))?L₂ (0, T; H ^?1 (Ω)). Here, we make crucial use of this result to present the following marked improvement: the mapu → f ?f _t is continuous fromL ₂ (0, T; L ₂ (Γ)) intoC([0, T]; L ₂ (Ω))?C([0, T]; H ^?1 (Ω)). Our approach uses the cosine operator model introduced in [6], along with crucial relevant fact from cosine operator theory. A new trace theory result, on which we base our proof here, plays also a decisive role in the corresponding quadratic optimal control problem [7]. Whenu, instead, acts in the Neumann B. C. and Ω is either a sphere or a parallelepiped, the same approach leads to the same improvement over results obtained in [6] to the regularity int of the solution (i.e., fromL ₂ (0, T) toC[0, T]).     

Uniqueness Theorems for Some Free Boundary Problems in Univalent Functions

Let Φ be a bounded positive continuous function in the complex plane C. We will consider the following problem. We can find a simply connected region Ω containing the origin such that the Riemann mapping function f _Ω mapping the unit disk D onto Ω will satisfy <artwork name="GCOV31013eu1"> We show that there is a unique solution which is starlike with respect to the origin when Φ satisfies <artwork name="GCOV31013eu2"> We also show that if in addtion, ?Φ/?|w|>0 and ?log Φ is subharmonic, then Ω will be a convex region.     

Extinction and non-extinction for a polytropic filtration equation with a nonlocal source

In this article, the authors establish the conditions for the extinction of solutions, in finite time, of the fast diffusive polytropic filtration equation u _t ?=?div(|?u ^m | ^p?2?u ^m )?+?a∫_Ω u ^q (y,?t)dy with a, q, m?>?0, p?>?1, m(p???1)?R ^N (N?>?2). More precisely speaking, it is shown that if q?>?m(p???1), any non-negative solution with small initial data vanishes in finite time, and if 0?q?m(p???1), there exists a solution which is positive in Ω for all t?>?0. For the critical case q?=?m(p???1), whether the solutions vanish in finite time or not depends on the comparison between a and μ, where μ?=?∫?_Ωφ ^p?1(x)dx and φ is the unique positive solution of the elliptic problem ?div(|?φ| ^p?2?φ)?=?1, x?∈?Ω; φ(x)?=?0, x?∈??Ω.     

Solutions with boundary‐layers and spike‐layers to singularly perturbed quasilinear Dirichlet problems

The structure of nontrivial nonnegative solutions to singularly perturbed quasilinear Dirichlet problems of the form –?Δ_pu = f(u) in Ω, u = 0 on ?Ω, Ω ? R ^N a bounded smooth domain, is studied as ? → 0⁺, for a class of nonlinearities f(u) satisfying f(0) = f(z₁) = f(z₂) = 0 with 0 < z₁ < z₂, f < 0 in (0, z₁), f > 0 in (z₁, z₂) and f(u)/u^p–1 = –∞. It is shown that there are many nontrivial nonnegative solutions with spike‐layers. Moreover, the measure of each spike‐layer is estimated as ? → 0⁺. These results are applied to the study of the structure of positive solutions of the same problems with f changing sign many times in (0,∞). Uniqueness of a solution with a boundary‐layer and many positive intermediate solutions with spike‐layers are obtained for ? sufficiently small. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Blow-Up of Positive Solutions for A Riharmonic Equation Involving Nearly Critical Exponent

In this paper a biharmonic problem with Navier boundary condition involving nearly critical growth is considered: △²=u^{(n+4)/(n-4)-r} u > 0 inΩ and u=△u=0 on ?Ω, where iΩs a bounded smooth convex domain in Rⁿ (n≥5) and r > 0 is small. We show that any sequence of positive solutions with r→0 has to blow up and concentrate at finitely many points in the interior of the domain ω. With blow-up argument, we also give the energy a priori estimate of positive solutions.     

The composition operators on weighted Bloch space

We will characterize the boundedness and compactness of the composition operators on weighted Bloch space B _log = { f ? H(D): sup_{z ? D} (1-| z|²) ( log\frac21-| z|² )| f￠(z)| B_{ \log }= \{ f \in H(D): \sup_{z \in D } (1-\left| z\right|^2) \left( \log \frac{2}{1-\left| z\right|^2} \right)\left| f'(z)\right| < +￥} +\infty \} , where H(D) be the class of all analytic functions on D.     

A Remark on the Sobolev Regularity of Classical Solutions of Strongly Elliptic Equations

We prove that C^2,α(Ω ) solutions of problem (1.2) below are in H^m+2(Ω) for all m ∈ ?, if f and the coefficients are in H^m (Ω) n C^0,α (Ω ) Previously, this result was explicitly known only if m> n/2 (or if m = 0). A similar result holds for the quasi-linear equation (1.11) below.