Integral properties of the classical warping function of a simply connected domain

Let u(z,G) be the classical warping function of a simply connected domain G. We prove that the L ^p -norms of the warping function with different exponents are related by a sharp isoperimetric inequality, including the functional u(G) = sup _x∈G u(x, G). A particular case of our result is the classical Payne inequality for the torsional rigidity of a domain. On the basis of the warping function, we construct a new physical functional possessing the isoperimetric monotonicity property. For a class of integrals depending on the warping function, we also obtain a priori estimates in terms of the L ^p -norms of the warping function as well as the functional u(G). In the proof, we use the estimation technique on level lines proposed by Payne.     

Monotonicity,continuity and differentiability results for the Lp Hardy constant

We consider the L^p Hardy inequality involving the distance to the boundary for a domain in the n-dimensional Euclidean space. We study the dependence on p of the corresponding best constant and we prove monotonicity, continuity and differentiability results. The focus is on non-convex domains in which case such a constant is in general not explicitly known.     

Isoperimetric inequalities for L p -norms of the stress function of a multiply connected plane domain

Let u(x,G) be the stress function of a multiply connected plane domain G. We construct new domain functionals depending on this stress function which are isoperimetrically monotone with respect to the free parameter. A particular case of the proved result is the Payne inequality for the torsional rigidity of G.     

Strichartz type estimates and the well-posedness of an energy critical 2D wave equation in a bounded domain

We study the well-posedness of the Cauchy problem with Dirichlet or Neumann boundary conditions associated to an H¹-critical semilinear wave equation on a smooth bounded domain Ω⊂R². First, we prove an appropriate Strichartz type estimate using the Lq spectral projector estimates of the Laplace operator. Our proof follows Burq, Lebeau and Planchon (2008) [4]. Then, we show the global well-posedness when the energy is below or at the threshold given by the sharp Moser-Trudinger inequality. Finally, in the supercritical case, we prove an instability result using the finite speed of propagation and a quantitative study of the associated ODE with oscillatory data.     

Pointwise inequalities of logarithmic type in Hardy-Hölder spaces

We prove some optimal logarithmic estimates in the Hardy space H ^∞(G) with Hölder regularity, where G is the open unit disk or an annular domain of ?. These estimates extend the results established by S.Chaabane and I.Feki in the Hardy-Sobolev space H ^k,∞ of the unit disk and those of I. Feki in the case of an annular domain. The proofs are based on a variant of Hardy-Landau-Littlewood inequality for Hölder functions. As an application of these estimates, we study the stability of both the Cauchy problem for the Laplace operator and the Robin inverse problem.     

Isoperimetric monotony of the L p -norm of the warping function of a plane simply connected domain

Let G be a simply connected domain and let u(x,G) be its warping function. We prove that L ^p -norms of functions u and u ^?1 are monotone with respect to the parameter p. This monotony also gives isoperimetric inequalities for norms that correspond to different values of the parameter p. The main result of this paper is a generalization of classical isoperimetric inequalities of St.Venant-Pólya and the Payne inequalities.     

A Payne�CRayner Type Inequality for the Robin Problem on Arbitrary Minimal Surfaces in {\mathbb{R}^N}

We prove a Payne?CRayner type inequality for the first eigenfunction of the Laplacian with Robin boundary condition on any compact minimal surface with boundary in ${\mathbb{R}^N}$ . We emphasize that no topological condition is necessary on the boundary.     

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.     

Gårding’s inequality for elliptic operators with degeneracy

In this paper, we prove Gårding’s weighted inequality for degenerate elliptic operators in an arbitrary (bounded or unbounded) domain of n-dimensional Euclidean space ? ⁿ and use this inequality to study the unique solvability of a specific variational problem. It is assumed that the lower coefficients of the operators under consideration belong to some weighted L _p -spaces.     

Weighted integral inequalities for conjugate A-harmonic tensors

We first prove a local weighted integral inequality for conjugate A-harmonic tensors. Then, as an application of our local result, we prove a global weighted integral inequality for conjugate A-harmonic tensors in L^s(μ)-averaging domains, which can be considered as a generalization of the classical result. Finally, we give applications of the above results to quasiregular mappings.     

A Hardy inequality in the half-space

Here we prove a Hardy-type inequality in the upper half-space which generalize an inequality originally proved by Maz’ya (Sobolev Spaces, Springer, Berlin, 1985, p. 99). Here we present a different proof, which enable us to improve the constant in front of the remainder term. We will also generalize the inequality to the L^p case.     

L 1 Hardy Inequalities with Weights

We prove sharp homogeneous improvements to L ¹ weighted Hardy inequalities involving distance from the boundary. In the case of a smooth domain, we obtain lower and upper estimates for the best constant of the remainder term. These estimates are sharp in the sense that they coincide when the domain is a ball or an infinite strip. In the case of a ball, we also obtain further improvements.     

The general optimal L-Euclidean logarithmic Sobolev inequality by Hamilton-Jacobi equations

We prove a general optimal L^p-Euclidean logarithmic Sobolev inequality by using Prékopa-Leindler inequality and a special Hamilton-Jacobi equation. In particular we generalize the inequality proved by Del Pino and Dolbeault in (J. Funt. Anal.).     

On the Stability of the p-Affine Isoperimetric Inequality

Employing the affine normal flow, we prove a stability version of the p-affine isoperimetric inequality for p≥1 in ?² in the class of origin-symmetric convex bodies. That is, if K is an origin-symmetric convex body in ?² such that it has area π and its p-affine perimeter is close enough to the one of an ellipse with the same area, then, after applying a special linear transformation, K is close to an ellipse in the Hausdorff distance.     

Extension of two theorems of Payne to some nonlinear Dirichlet problems

In this paper we prove that under certain convexity and symmetry assumptions on a domain in the plane any positive solutionu, of Δu+λf(u)=0, in,D,u=0 on ?D has only one interior critical point. This extends results of L. E. Payne [1].     

Square area integral estimates for subharmonic functions in NTA domains

In this paper, we prove a good-λ inequality between the nontangential maximal function and the square area integral of a subharmonic functionu in a bounded NTA domainD inR ⁿ . We achieve this by showing that a weighted Riesz measure ofu is a Carleson measure, with the Carleson norm bounded by a constant independent ofu. As consequences of the good-λ inequality, we obtain McConnell-Uchiyama's inequality and an analogue of Murai-Uchiyama's inequality for subharmonic functions inD.     

A Brunn-Minkowski inequality for the Hessian eigenvalue in three-dimensional convex domain

We use the deformation methods to obtain the strictly log concavity of solution of a class Hessian equation in bounded convex domain in R³, as an application we get the Brunn-Minkowski inequality for the Hessian eigenvalue and characterize the equality case in bounded strictly convex domain in R³.     

Some remarks on a discrepancy in compact groups

In this paper we investigate a discrepancy and a L ² discrepancy on compact groups which were introduced by E. Hlawka and W. Fleischer. First we show that this L ² discrepancy is a generalization of the classical diaphony and can be expressed as a finite double sum. We also give estimations of quadrature errors for smooth functions. Then we prove an inequality of Erdos-Turán type for the discrepancy on compact abelian groups and study this inequality in the case of the torus and the dyadic group.     

Estimation of the L p -norms of stress functions for finitely connected plane domains

Let u(x, G) be the classical stress function of a finitely connected plane domain G. The isoperimetric properties of the L ^p -norms of u(x, G) are studied. Payne’s inequality for simply connected domains is generalized to finitely connected domains. It is proved that the L ^p -norms of the functions u(x, G) and u ^?1 (x, G) strictly decrease with respect to the parameter p, and a sharp bound for the rate of decrease of the L ^p -norms of these functions in terms of the corresponding L ^p -norms of the stress function for an annulus is obtained. A new integral inequality for the L ^p -norms of u(x, G), which is an analog of the inequality obtained by F. G. Avkhadiev and the author for the L ^p -norm of conformal radii, is proved.