Representations of Solutions of Laplacian Boundary Value Problems on Exterior Regions

This paper treats the well-posedness and representation of solutions of Poisson’s equation on exterior regions $U\subsetneq{\mathbb{R}}^{N}$ with N≥3. Solutions are sought in a space E ¹(U) of finite energy functions that decay at infinity. This space contains H ¹(U) and existence-uniqueness theorems are proved for the Dirichlet, Robin and Neumann problems using variational methods with natural conditions on the data. A decomposition result is used to reduce the problem to the evaluation of a standard potential and the solution of a harmonic boundary value problem. The exterior Steklov eigenproblems for the Laplacian on U are described. The exterior Steklov eigenfunctions are proved to generate an orthogonal basis for the subspace of harmonic functions and also of certain boundary trace spaces. Representations of solutions of the harmonic boundary value problem in terms of these bases are found, and estimates for the solutions are derived. When U is the region exterior to a 3-d ball, these Steklov representations reduce to the classical multi-pole expansions familiar in physics and engineering analysis.     

Bounds for the number of points on curves over finite fields

Let U be a bounded open subset of ?^d, d ≥ 2 and f ∈ C(?U). The Dirichlet solution f^CU of the Dirichlet problem associated with the Laplace equation with a boundary condition f is not continuous on the closure ū of U in general if U is not regular but it is always Baire-one.Let H(U) be the space of all functions continuous on the closure ū and harmonic on U and F(H(U)) be the space of uniformly bounded absolutely convergent series of functions in H(U). We prove that f^CU can be obtained as a uniform limit of a sequence of functions in F(H(U)). Thus f^CU belongs to the subclass B_1/2 of Baire-one functions studied for example in [8]. This is not only an improvement of the result obtained in [10] but it also shows that the Dirichlet solution on the closure ū can share better properties than to be only a Baire-one function. Moreover, our proof is more elementary than that in [10].A generalization to the abstract context of simplicial function space on a metrizable compact space is provided.We conclude the paper with a brief discussion on the solvability of the abstract Dirichlet problem with a boundary condition belonging to the space of differences of bounded semicontinuous functions complementing the results obtained in [17].     

Generalized Harmonic Functions and the Dewetting of Thin Films

This paper describes the solvability of Dirichlet problems for Laplace's equation when the boundary data is not smooth enough for the existence of a weak solution in H¹Ω. Scales of spaces of harmonic functions and of boundary traces are defined and the solutions are characterized as limits of classical harmonic functions in special norms. The generalized harmonic functions, and their norms, are defined using series expansions involving harmonic Steklov eigenfunctions on the domain. It is shown that the usual trace operator has a continuous extension to an isometric isomorphism of specific spaces. This provides a characterization of the generalized solutions of harmonic Dirichlet problems. Numerical simulations of a model problem are described. This problem is related to the dewetting of thin films and the associated phenomenology is described.     

The class of solenoidal planar-helical vector fields

The class of solenoidal vector fields whose lines lie in planes parallel to R ² is constructed by the method of mappings. This class exhausts the set of all smooth planarhelical solutions of Gromeka’s problem in some domain D ? R ³. In the case of domains D with cylindrical boundaries whose generators are orthogonal to R ², it is shown that the choice of a specific solution from the constructed class is reduced to the Dirichlet problem with respect to two functions that are harmonic conjugates in D ² = D∩R ²; i.e., Gromeka’s nonlinear problem is reduced to linear boundary value problems. As an example, a specific solution of the problem for an axisymmetric layer is presented. The solution is based on solving Dirichlet problems in the form of series uniformly convergent in \(\bar D^2\) in terms of wavelet systems that form bases of various spaces of functions harmonic in D ².     

Positive solutions of nonlinear elliptic boundary value problems

This paper considers the problem of finding positive vector-valued solutions U of the nonlinear elliptic boundary value problem L(U) + f(x, U) = 0 on a bounded region Ω, $\text{U +}\text{?U}\text{?v}\text{= 0}$ on ?Ω. The operator L is uniformly elliptic and in divergence form, and f is, roughly speaking, superlinear; by the positivity of U is meant the positivity of each component of U on Ω. Under certain growth conditions on f and some further technical assumptions, the existence of a positive solution is proved, an a priori bound on all positive solutions is obtained, and a certain fixed point index is proved equal to ? 1. As an example, information about fixed point indices is used to allow perturbations of the form ?h(x, U, DU). In the final section, an essentially best possible theorem is given for Ω a ball and for radially symmetric solutions of the Laplacian with Dirichlet boundary conditions.     

Enveloppes polynômiales d'ensembles compacts invariants

We consider the following problem: let V^? be a finite dimensional vector space, and U be a compact group of ?‐linear automorphisms of V^?. The polynomial envelope of a compact set Q ? V^? is defined as where ??(V^?) denotes the space of holomorphic polynomial functions on V^?. The problem is to determine the polynomial envelope of a compact set which is U‐invariant. We solve the problem when U is the isotropy subgroup at the origin of the automorphism group of a bounded symmetric domain of tube type. The case of a domain of type II has been solved by C. Sacré [1992], and, for a domain of type IV, it has been solved by L. Bou Attour [1993]. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Monogenic functions in the biharmonic boundary value problem

We consider a commutative algebra over the field of complex numbers with a basis {e₁,e₂} satisfying the conditions , . Let D be a bounded domain in the Cartesian plane xOy and D_ζ={xe₁+ye₂:(x,y)∈D}. Components of every monogenic function Φ(xe₁+ye₂) = U₁(x,y)e₁+U₂(x,y)ie₁+U₃(x,y)e₂+U₄(x,y)ie₂ having the classic derivative in D_ζ are biharmonic functions in D, that is, Δ²U_j(x,y) = 0 for j = 1,2,3,4. We consider a Schwarz‐type boundary value problem for monogenic functions in a simply connected domain D_ζ. This problem is associated with the following biharmonic problem: to find a biharmonic function V(x,y) in the domain D when boundary values of its partial derivatives ?V/?x, ?V/?y are given on the boundary ?D. Using a hypercomplex analog of the Cauchy‐type integral, we reduce the mentioned Schwarz‐type boundary value problem to a system of integral equations on the real axes and establish sufficient conditions under which this system has the Fredholm property. Copyright © 2015 John Wiley & Sons, Ltd.     

Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains

For D, a bounded Lipschitz domain in Rⁿ, n ? 2, the classical layer potentials for Laplace's equation are shown to be invertible operators on L²(?D) and various subspaces of L²(?D). For 1 < p ? 2 and data in L^p(?D) with first derivatives in L^p(?D) it is shown that there exists a unique harmonic function, u, that solves the Dirichlet problem for the given data and such that the nontangential maximal function of ▽u is in L^p(?D). When n = 2 the question of the invertibility of the layer potentials on every L^p(?D), 1 < p < ∞, is answered.     

Spectral deformations of one-dimensional Schrödinger operators

We provide a complete spectral characterization of a new method of constructing isospectral (in fact, unitary) deformations of general Schrödinger operatorsH=?d ²/dx ²+V in $H = - d^2 /dx^2 + V in \mathcal{L}^2 (\mathbb{R})$ . Our technique is connected to Dirichlet data, that is, the spectrum of the operatorH ^D onL ²((?∞,x ₀)) ⊕L ²((x ₀, ∞)) with a Dirichlet boundary condition atx ₀. The transformation moves a single eigenvalue ofH ^D and perhaps flips which side ofx ₀ the eigenvalue lives. On the remainder of the spectrum, the transformation is realized by a unitary operator. For cases such asV(x)→∞ as |x|→∞, whereV is uniquely determined by the spectrum ofH and the Dirichlet data, our result implies that the specific Dirichlet data allowed are determined only by the asymptotics asE→∞.     

Construction of Carleman formulas by using mixed problems with parameter-dependent boundary conditions

Let D be an open connected subset of the complex plane C with sufficiently smooth boundary ?D. Perturbing the Cauchy problem for the Cauchy–Riemann system ??u = f in D with boundary data on a closed subset S ? ?D, we obtain a family of mixed problems of the Zaremba-type for the Laplace equation depending on a small parameter ε ∈ (0, 1] in the boundary condition. Despite the fact that the mixed problems include noncoercive boundary conditions on ?D\S, each of them has a unique solution in some appropriate Hilbert space H ⁺(D) densely embedded in the Lebesgue space L ₂(?D) and the Sobolev–Slobodetski? space H ^1/2?δ(D) for every δ > 0. The corresponding family of the solutions {u _ε} converges to a solution to the Cauchy problem in H ⁺(D) (if the latter exists). Moreover, the existence of a solution to the Cauchy problem in H ⁺(D) is equivalent to boundedness of the family {u _ε} in this space. Thus, we propose solvability conditions for the Cauchy problem and an effective method of constructing a solution in the form of Carleman-type formulas.     

Logarithmic Bounds on Sobolev Norms for Time Dependent Linear Schrödinger Equations

We consider the Aharonov–Bohm effect for the Schrödinger operator H = (?i? _x  ? A(x))² + V(x) and the related inverse problem in an exterior domain Ω in R ² with Dirichlet boundary condition. We study the structure and asymptotics of generalized eigenfunctions and show that the scattering operator determines the domain Ω and H up to gauge equivalence under the equal flux condition. We also show that the flux is determined by the scattering operator if the obstacle Ω ^c is convex.     

Inversion formulas for complex radon transform on projective varieties and boundary value problems for systems of linear PDEs

Let G ? ?P ⁿ be a linearly convex compact set with smooth boundary, D = ?P ⁿ \ G, and let D* ? (?P ⁿ )* be the dual domain. Then for an algebraic, not necessarily reduced, complete intersection subvariety V of dimension d we construct an explicit inversion formula for the complex Radon transform R _V : H ^d,d?1(V ∩ D) → H ^1,0(D*) and explicit formulas for solutions of an appropriate boundary value problem for the corresponding system of differential equations with constant coefficients on D*.     

Potential Theory of Special Subordinators and Subordinate Killed Stable Processes

In this paper we introduce a large class of subordinators called special subordinators and study their potential theory. Then we study the potential theory of processes obtained by subordinating a killed symmetric stable process in a bounded open set D with special subordinators. We establish a one-to-one correspondence between the nonnegative harmonic functions of the killed symmetric stable process and the nonnegative harmonic functions of the subordinate killed symmetric stable process. We show that nonnegative harmonic functions of the subordinate killed symmetric stable process are continuous and satisfy a Harnack inequality. We then show that, when D is a bounded κ-fat set, both the Martin boundary and the minimal Martin boundary of the subordinate killed symmetric stable process in D coincide with the Euclidean boundary ∂D. The research of this author is supported in part by MZOS grant 0037107 of the Republic of Croatia and in part by a joint US-Croatia grant INT 0302167.     

Harmonic maps with singular boundary value from complex hyperbolic spaces into rank one symmetric spaces

In this paper we consider the Dirichlet problem at infinity of proper harmonic maps from noncompact complex hyperbolic space to a rank one symmetric space N of noncompact type with singular boundary data . Under some conditions on f, we show that the Dirichlet problem at infinity admits a harmonic map which assumes the boundary data f continuously. Received: March 11, 1999 / Accepted April 23, 1999     

Boundary value problems in generalized biaxially symmetric potential theory

Interior boundary value problems are solved for the operator of generalized biaxially symmetric potential theory. The boundary conditions consist of Dirichlet data on the nonsingular part of the boundary and Dirichlet data or growth restrictions on the singular hyperplanes, depending on the values of parameters of the operator. Continuation of solutions beyond the singular hyperplanes is considered, yielding an improvement of a result of Huber. Potential theoretic methods are used for the investigation.     

The Cauchy problem in Sobolev spaces for Dirac operators

In this paper we consider the Cauchy problem as a typical example of ill-posed boundary-value problems. We obtain the necessary and (separately) sufficient conditions for the solvability of the Cauchy problem for a Dirac operator A in Sobolev spaces in a bounded domain D ? ? ⁿ with a piecewise smooth boundary. Namely, we reduce the Cauchy problem for the Dirac operator to the problem of harmonic extension from a smaller domain to a larger one. Moreover, along with the solvability conditions for the problem, using bases with double orthogonality, we construct a Carleman formula for recovering a function u in a Sobolev space H ^s (D), s ∈ ?, from its values on Γ and values Au in D, where Γ is an open connected subset of the boundary ?D. It is worth pointing out that we impose no assumptions about geometric properties of the domain D, except for its connectedness.     

Conformal invariants associated with quadratic differentials

We associate a functional of pairs of simply-connected regions D₂ ? D₁ to any quadratic differential on D₁ with specified singularities. This functional is conformally invariant, monotonic, and negative. Equality holds if and only if the inner domain is the outer domain minus trajectories of the quadratic differential. This generalizes the simply-connected case of results of Z. Nehari [20], who developed a general technique for obtaining inequalities for conformal maps and domain functions from contour integrals and the Dirichlet principle for harmonic functions. Nehari’s method corresponds to the special case that the quadratic differential is of the form (?q)² for a singular harmonic function q on D₁.As an application we give a one-parameter family of monotonic, conformally invariant functionals which correspond to growth theorems for bounded univalent functions. These generalize and interpolate the Pick growth theorems, which appear in a conformally invariant form equivalent to a two-point distortion theorem of W. Ma and D. Minda [16].     

An algorithm for Gauss harmonic formulas

LetD be an open, bounded, simply-connected region inR ² with boundaryB. Let (x*,y*) be an arbitrary point ofD. This paper constructs an algorithm for computing Gauss harmonic formulas forD and the point (x*,y*). Such formulas approximate a harmonic function at (x*,y*) in terms of a linear combination of its boundary values. Such formulas are useful for approximating the solution of the Dirichlet problem, especially when the problem is to be solved many times at the same point with different boundary values.     

Harmonic functions of subordinate killed Brownian motion

In this paper we study harmonic functions of subordinate killed Brownian motion in a domain D. We first prove that, when the killed Brownian semigroup in D is intrinsic ultracontractive, all nonnegative harmonic functions of the subordinate killed Brownian motion in D are continuous and then we establish a Harnack inequality for these harmonic functions. We then show that, when D is a bounded Lipschitz domain, both the Martin boundary and the minimal Martin boundary of the subordinate killed Brownian motion in D coincide with the Euclidean boundary ∂D. We also show that, when D is a bounded Lipschitz domain, a boundary Harnack principle holds for positive harmonic functions of the subordinate killed Brownian motion in D.