Sharp estimates of Hardy constants for domains with special boundary properties

We investigate the behavior of Hardy constants in domains whose boundaries have at least one regular point.     

Construction of envelopes of holomorphy for some classes of special domains

The envelope of holomorphy E(Ω) of any domain Ω in Cⁿ which is either of circular type or a tube, is constructed in terms of the envelope of a lower dimensional complex space. As consequences, conditions for the univalence of E(Ω) are proved. The research was done primarily while the first author was at the Dipartimento di Matematica of the Università di Trento.     

The third boundary value problem in potential theory for domains with a piecewise smooth boundary

The paper investigates the third boundary value problem for the Laplace equation by the means of the potential theory. The solution is sought in the form of the Newtonian potential (1), (2), where is the unknown signed measure on the boundary. The boundary condition (4) is weakly characterized by a signed measure the corresponding operator on the space of signed measures on the boundary of the investigated domain G. If there is 0 such that the essential spectral radius of is smaller than || (for example, if G R ³ is a domain with a piecewise smooth boundary and the restriction of the Newtonian potential on G is a finite continuous functions) then the third problem is uniquely solvable in the form of a single layer potential (1) with the only exception which occurs if we study the Neumann problem for a bounded domain. In this case the problem is solvable for the boundary condition for which (G) = 0.     

Uniqueness classes of solutions of boundary problems for nondivergent second order parabolic equations in noncylindrical domains

In an unbounded noncylindrical domain G we consider classical solutions of general second order linear parabolic equations satisfying a Dirichlet boundary condition on the parabolic part of the boundary of G. On the basis of the maximum principle we single out classes of uniqueness of such solutions depending on the geometry of the domain G which are analogs of the classes of A. N. Tikhonov and S. Täcklind.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 7, pp. 924–930, July, 1990.     

Geometric and transformational properties of Lipschitz domains,Semmes-Kenig-Toro domains,and other classes of finite perimeter domains

In the first part of this article we give intrinsic characterizations of the classes of Lipschitz and C¹ domains. Under some mild, necessary, background hypotheses (of topological and geometric measure theoretic nature), we show that a domain is Lipschitz if and only if it has a continuous transversal vector field. We also show that if the geometric measure theoretic unit normal of the domain is continuous, then the domain in question is of class C¹. In the second part of the article, we study the invariance of various classes of domains of locally finite perimeter under bi-Lipschitz and C¹ diffeomorphisms of the Euclidean space. In particular, we prove that the class of bounded regular SKT domains (previously called chord-arc domains with vanishing constant, in the literature) is stable under C¹ diffeomorphisms. A number of other applications are also presented. Acknowledgements and Notes. The work of the authors was supported in part by NSF grants DMS-0245401, DMS-0653180, DMS-FRG0456306, and DMS-0456861.     

On domains of regularity of the solutions of some special classes of Monge-Ampère-type equations

We present several results on estimates from above of the regularity domains for special classes of solutions to Monge-Ampère-type equations (in particular, periodic at least in one variable). Also, we present some geometrical and geophysical applications. Namely, we are concerned with the problem on dimensions of a single-valued projection on a plane of a surface with separated-from-zero negative Gaussian curvature and discuss the existence or nonexistence of a solution to the pressure-wind balance equation on a torus. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 1, pp. 237–246, 2006.     

Comparison of potential theoretic properties of rough domains

We discuss the relationships between the notions of intrinsic ultracontractivity, the parabolic Harnack principle, compactness of the 1-resolvent of the Neumann Laplacian, and the non-trap property for Euclidean domains with finite Lebesgue measure. In particular, we give an answer to an open problem raised by Davies and Simon in 1984 about the possible relationship between intrinsic ultracontractivity for the Dirichlet Laplacian in a domain and compactness of the 1-resolvent of the Neumann Laplacian in .

     

Extremal properties of solutions of special classes of a hyperbolic-type equation

Special classes of solutions of the Cauchy problem for a hyperbolic equation are introduced and local extremum principles for the cases of positive and negative values of the parameter of the equation are established.     

Isoperimetric inequalities for special classes of curves

In this paper the classical Banchoff-Pohl inequality, an isoperimetric inequality for nonsimple closed curves in the Euclidean plane, involving the square of the winding number, is sharpened for homothetic or Abresch-Langer solutions of curve shortening. For a larger class of curves and for rotationally symmetric curves, further isoperimetric inequalities containing the rotation number and the winding number, are presented.     

Unique continuation on the boundary for Dini domains

We show that the normal derivative of a harmonic function which vanishes on an open subset of the boundary of a Dini domain cannot vanish on a subset of positive surface measure.

     

Numerical implementation of the EDEM for modified Helmholtz BVPs on annular domains

In a recent paper by the current authors a new methodology called the Extended-Domain-Eigenfunction-Method (EDEM) was proposed for solving elliptic boundary value problems on annular-like domains. In this paper we present and investigate one possible numerical algorithm to implement the EDEM. This algorithm is used to solve modified Helmholtz BVPs on annular-like domains. Two examples of annular-like domains are studied. The results and performance are compared with those of the well-known boundary element method (BEM). The high accuracy of the EDEM solutions and the superior efficiency of the EDEM over the BEM, make EDEM an excellent alternate candidate to use in the animation industry, where speed is a predominant requirement, and by the scientific community where accuracy is the paramount objective.     

Convolutions for special classes of harmonic univalent functions

Ruscheweyh and Sheil-Small proved the PólyarSchoenberg conjecture that the class of convex analytic functions is closed under convolution or Hadamard product. They also showed that close-to-convexity is preserved under convolution with convex analytic functions. In this note, we investigate harmonic analogs. Beginning with convex analytic functions, we form certain harmonic functions which preserve close-to-convexity under convolution. An auxiliary function enables us to obtain necessary and sufficient convolution conditions for convex and starlike harmonic functions, which lead to sufficient coefficient bounds for inclusion in these classes.     

Qualitative properties of nodal solutions of semilinear elliptic equations in radially symmetric domains

We study the qualitative properties of sign changing solutions of the Dirichlet problem $\mathrm{\Delta }u+f(u)=0$ in Ω, $u=0$ on ?Ω, where Ω is a ball or an annulus and f is a $C^1$ function with $f(0)?0$. We prove that any radial sign changing solution has a Morse index bigger or equal to $N+1$ and give sufficient conditions for the nodal surface of a solution to intersect the boundary. In particular, we prove that any least energy nodal solution is non radial and its nodal surface touches the boundary. To cite this article: A. Aftalion, F. Pacella, C. R. Acad. Sci. Paris, Ser. I 339 (2004).