The best Sobolev trace constant in a domain with oscillating boundary

In this paper we study homogenization problems for the best constant for the Sobolev trace embedding W^1,p(Ω)?L^q(∂Ω)$W^{1,p}\left(\Omega \right)?L^q\left(\partial \Omega \right)$ in a bounded smooth domain when the boundary is perturbed by adding an oscillation. We find that there exists a critical size of the amplitude of the oscillations for which the limit problem has a weight on the boundary. For sizes larger than critical the best trace constant goes to zero and for sizes smaller than critical it converges to the best constant in the domain without perturbations.     

Optimal boundary holes for the Sobolev trace constant

In this paper we study the problem of minimizing the Sobolev trace Rayleigh quotient among functions that vanish in a set contained on the boundary ∂Ω of given boundary measure.We prove existence of extremals for this problem, and analyze some particular cases where information about the location of the optimal boundary set can be given. Moreover, we further study the shape derivative of the Sobolev trace constant under regular perturbations of the boundary set.     

Weighted Sobolev spaces with applications to singular nonlinear boundary value problems

Variational methods are used in a weighted Sobolev space to prove the existence of solutions for a certain class of singular nonlinear ordinary differential equations.     

Boundary trace embedding theorems for variable exponent Sobolev spaces

We study boundary trace embedding theorems for variable exponent Sobolev space W1,p(⋅)(Ω). Let Ω be an open (bounded or unbounded) domain in RN satisfying strong local Lipschitz condition. Under the hypotheses that p∈L^∞(Ω), 1?infp(x)?supp(x)<N, |∇p|∈Lγ(⋅)(Ω), where γ∈L^∞(Ω) and infγ(x)>N, we prove that there is a continuous boundary trace embedding W1,p(⋅)(Ω)→Lq(⋅)(∂Ω) provided q(⋅), a measurable function on ∂Ω, satisfies condition for x∈∂Ω.     

Sobolev functions whose inner trace at the boundary is zero

Let Ω⊂R ⁿ be an arbitrary open set. In this paper it is shown that if a Sobolev functionf∈W ^1,p (Ω) possesses a zero trace (in the sense of Lebesgue points) on ϖΩ, thenf is weakly zero on ϖΩ in the sense thatf∈W ₀ ^1,p (Ω).     

The effect of boundary geometry on Hankel operators belonging to the trace ideals of Bergman spaces

The characterization of thosef for which the Hankel operatorsH _f belongs to various trace ideals over Bergman spaces on pseudoconvex domains of finite type in complex dimension two is given. In particular, we determine how the cutoff values are affected by the boundary geometry.All three authors supported by grants from the National Science Foundation     

A semilinear non-classical pseudodifferential boundary value problem in the Sobolev spaces

We study the solvability of a semilinear non-classical pseudodifferential boundary value problem in the Sobolev spaces H_l,p,q, 1<p<∞, depending on a complex parameter q. To cite this article: Y.V. Egorov et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

The density of infinitely differentiable functions in Sobolev spaces with mixed boundary conditions

We present a detailed proof of the density of the set in the space of test functions V ⊂ H ¹ (Ω) that vanish on some part of the boundary ∂Ω of a bounded domain Ω. This work was supported by the grants GAČR 201/03/0570 and MSM 262100001.     

Elliptic equations with nonzero boundary conditions in weighted Sobolev spaces

We present weighted Sobolev spaces along with a trace theorem and an interpolation theorem for the spaces. Then we solve nonzero boundary value problems for elliptic equations in .     

Holomorphic Sobolev spaces associated to compact symmetric spaces

Using Gutzmer's formula, due to Lassalle, we characterise the images of Sobolev spaces under the Segal-Bargmann transform on compact Riemannian symmetric spaces. We also obtain necessary and sufficient conditions on a holomorphic function to be in the image of smooth functions and distributions under the Segal-Bargmann transform.     

On the Sobolev trace Theorem for variable exponent spaces in the critical range

In this paper, we study the Sobolev trace Theorem for variable exponent spaces with critical exponents. We find conditions on the best constant in order to guaranty the existence of extremals. Then, we give local conditions on the exponents and on the domain (in the spirit of Adimurthy and Yadava) in order to satisfy such conditions and therefore to ensure the existence of extremals.     

Sobolev spaces associated to the harmonic oscillator

We define the Hermite-Sobolev spaces naturally associated to the harmonic oscillatorH = −δ+|x|². Structural properties, relations with the classical Sobolev spaces, boundedness of operators and almost everywhere convergence of solutions of the Schrodinger equation are also considered.     

On the boundary limits of monotone Sobolev functions in variable exponent Orlicz spaces

Our aim in this note is to deal with boundary limits of monotone Sobolev functions with ▽u∈ Lp(·)logLq(·)(B) for the unit ball BRn. Here p(·) and q(·) are variable exponents satisfyingthe log-Hlder and the log log-Hlder conditions, respectively.     

Sobolev inequalities on homogeneous spaces

We consider a homogeneous spaceX=(X, d, m) of dimension 1 and a local regular Dirichlet forma inL ² (X, m). We prove that if a Poincaré inequality of exponent 1p< holds on every pseudo-ballB(x, R) ofX, then Sobolev and Nash inequalities of any exponentq[p, ), as well as Poincaré inequalities of any exponentq[p, +), also hold onB(x, R).Lavoro eseguito nell'ambito del Contratto CNR Strutture variazionali irregolari.     

Real-valued valuations on Sobolev spaces

Continuous, SL(n) and translation invariant real-valued valuations on Sobolev spaces are classified.The centro-affine Hadwiger’s theorem is applied. In the homogeneous case, these valuations turn out to be L~p-norms raised to p-th power(up to suitable multipication scales).