Existence of minimizers of the Mumford-Shah functional with singular operators and unbounded data

We consider the regularization of linear inverse problems by means of the minimization of a functional formed by a term of discrepancy to data and a Mumford-Shah functional term. The discrepancy term penalizes the L ² distance between a datum and a version of the unknown function which is filtered by means of a non-invertible linear operator. Depending on the type of the involved operator, the resulting variational problem has had several applications: image deblurring, or inverse source problems in the case of compact operators, and image inpainting in the case of suitable local operators, as well as the modeling of propagation of fracture. We present counterexamples showing that, despite this regularization, the problem is actually in general ill-posed. We provide, however, existence results of minimizers in a reasonable class of smooth functions out of piecewise Lipschitz discontinuity sets in two dimensions. The compactness arguments we developed to derive the existence results stem from geometrical and regularity properties of domains, interpolation inequalities, and classical compactness arguments in Sobolev spaces.     

On an Inverse Problem for a Parabolic Equation

In this paper, we study the inverse problem of the reconstruction of the right-hand side of special form for a parabolic equation in u in which the coefficients of u _t and u depend on u _(x,t) , with overdetermination given by the integral of the solution over time. The Fredholm property for this problem and the existence and uniqueness theorems in Sobolev spaces are established.     

Bessel Potentials in Ahlfors Regular Metric Spaces

In this paper we introduce Bessel potentials and the Sobolev potential spaces resulting from them in the context of Ahlfors regular metric spaces. The Bessel kernel is defined using a Coifman type approximation of the identity, and we show integration against it improves the regularity of Lipschitz, Besov and Sobolev-type functions. For potential spaces, we prove density of Lipschitz functions, and several embedding results, including Sobolev-type embedding theorems. Finally, using singular integrals techniques such as the T1 theorem, we find that for small orders of regularity Bessel potentials are inversible, its inverse in terms of the fractional derivative, and show a way to characterize potential spaces, concluding that a function belongs to the Sobolev potential space if and only if itself and its fractional derivative are in L^p. Moreover, this characterization allows us to prove these spaces in fact coincide with the classical potential Sobolev spaces in the Euclidean case.     

A note on well-posedness for Camassa-Holm equation

In this note, we investigate the problem of well-posedness for a shallow water equation with data having critical regularity. Our results are based on the use of Besov spaces B_2,r^s (which generalize the Sobolev spaces H^s) with critical index s=3/2.     

Regularity of the inverse of a homeomorphism of a Sobolev–Orlicz space

Given a homeomorphism ? ∈ W _M ¹ , we determine the conditions that guarantee the belonging of the inverse of ? in some Sobolev–Orlicz space W _F ¹ . We also obtain necessary and sufficient conditions under which a homeomorphism of domains in a Euclidean space induces the bounded composition operator of Sobolev–Orlicz spaces defined by a special class of N-functions. Using these results, we establish requirements on a mapping under which the inverse homeomorphism also induces the bounded composition operator of another pair of Sobolev–Orlicz spaces which is defined by the first pair.     

Nonlinear Neumann–Transmission Problems for Stokes and Brinkman Equations on Euclidean Lipschitz Domains

The purpose of this paper is to use a layer potential analysis and the Leray–Schauder degree theory to show an existence result for a nonlinear Neumann–transmission problem corresponding to the Stokes and Brinkman operators on Euclidean Lipschitz domains with boundary data in L ^p spaces, Sobolev spaces, and also in Besov spaces.     

Carleman Estimates for the Schrdinger Equation and Applications to an Inverse Problem and an Observability Inequality

The authors prove Carleman estimates for the Schrdinger equation in Sobolev spaces of negative orders, and use these estimates to prove the uniqueness in the inverse problem of determining Lp-potentials. An L2-level observability inequality and unique continuation results for the Schrdinger equation are also obtained.     

The generalized solution of ill-posed boundary problem

In this paper, we define a kind of new Sobolev spaces, the relative Sobolev spaces Wk,p0(Ω,∑). Then an elliptic partial differential equation of the second order with an ill-posed boundary is discussed. By utilizing the ideal of the generalized inverse of an operator, we introduce the generalized solution of the ill-posed boundary problem. Eventually, the connection between the generalized inverse and the generalized solution is studied. In this way, the non-instability of the minimal normal least square solution of the ill-posed boundary problem is avoided.     

The Cauchy problem of generalized Landau-Lifshitz equation into S n

In this paper,we study the Cauchy problem of an integrable evolution system,i.e.,the n-dimensional generalization of third-order symmetry of the well-known Landau-Lifshitz equation.By rewriting this equation in a geometric form and applying the geometric energy method with a forth-order perturbation,we show the global well-posedness of the Cauchy problem in suitable Sobolev spaces.     

Finite elements for a prefractal transmission problem

In this Note we deal with the finite element approximation of a transmission problem across a prefractal curve approximating the von Koch fractal curve. We construct a mesh adapted to the geometric shape of the interface and we refine it consistently with some estimates in suitable weighted Sobolev spaces. In these spaces we also obtain an approximation error estimate. To cite this article: P. Bagnerini et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Analysis of a Large Eddy Simulation model based on anisotropic filtering

We consider a new Large Eddy Simulation model, derived with the introduction of suitable horizontal (anisotropic) differential filters. One main advantage of this filtering is that, for channel flows, there is no need for artificial boundary conditions. Hence, we can deal with some realistic problems, equipped with Dirichlet boundary conditions, in special bounded domains (at least those bounded only in one direction). Recent numerical results for a similar model, based on a derivation with wave-number asymptotics, are also recalled. After a detailed analysis of the properties of the differential filter, we prove that the resulting initial–boundary value problem is well-posed in suitable anisotropic Sobolev spaces, giving a strong mathematical support to the model we propose. Some remarks on higher-accuracy Approximate Deconvolution Models are also given in the last section.     

RECOVERY OF THE SINGULARITIES OF A POTENTIAL FROM FIXED ANGLE SCATTERING DATA

We study the inverse scattering problem for Schroedinger equation. We prove that for non-smooth potential the main singularities of the potential are contained in the Born approximation which can be obtained from measurement of the scattering amplitude in a single outgoing direction. We measure singularities in the scale of Sobolev spaces.     

Optimal Gaussian Sobolev embeddings

A reduction theorem is established, showing that any Sobolev inequality, involving arbitrary rearrangement-invariant norms with respect to the Gauss measure in Rn, is equivalent to a one-dimensional inequality, for a suitable Hardy-type operator, involving the same norms with respect to the standard Lebesgue measure on the unit interval. This result is exploited to provide a general characterization of optimal range and domain norms in Gaussian Sobolev inequalities. Applications to special instances yield optimal Gaussian Sobolev inequalities in Orlicz and Lorentz(-Zygmund) spaces, point out new phenomena, such as the existence of self-optimal spaces, and provide further insight into classical results.     

Mixed exterior Laplace's problem

In [C. Amrouche, V. Girault, J. Giroire, Dirichlet and Neumann exterior problems for the n-dimensional Laplace operator. An approach in weighted Sobolev spaces, J. Math. Pures Appl. 76 (1997) 55-81], authors study Dirichlet and Neumann problems for the Laplace operator in exterior domains of Rn. This paper extends this study to the resolution of a mixed exterior Laplace's problem. Here, we give existence, unicity and regularity results in Lp's theory with 1<p<∞, in weighted Sobolev spaces.     

Sobolev problem in the complete scale of Banach Spaces

In a bounded domainG ? ? ⁿ , whose boundary is the union of manifolds of different dimensions, we study the Sobolev problem for a properly elliptic expression of order 2m. The boundary conditions are given by linear differential expressions on manifolds of different dimensions. We study the Sobolev problem in the complete scale of Banach spaces. For this problem, we prove the theorem on a complete set of isomorphisms and indicate its applications.     

Two-dimensional vortex sheets for the nonisentropic Euler equations: Nonlinear stability

We show the short-time existence and nonlinear stability of vortex sheets for the nonisentropic compressible Euler equations in two spatial dimensions, based on the weakly linear stability result of Morando and Trebeschi (2008) [20]. The missing normal derivatives are compensated through the equations of the linearized vorticity and entropy when deriving higher-order energy estimates. The proof of the resolution for this nonlinear problem follows from certain a priori tame estimates on the effective linear problem in the usual Sobolev spaces and a suitable Nash–Moser iteration scheme.     

Approximation by polynomials and smooth functions in Sobolev spaces with respect to measures

The density of polynomials is straightforward to prove in Sobolev spaces W^k,p((a,b)), but there exist only partial results in weighted Sobolev spaces; here we improve some of these theorems. The situation is more complicated in infinite intervals, even for weighted L^p spaces; besides, in the present paper we have proved some other results for weighted Sobolev spaces in infinite intervals.     

Global existence and asymptotic behavior of solutions for a semi-linear wave equation

In this paper, we investigate the initial value problem for a semi-linear wave equation in n-dimensional space. Based on the decay estimate of solutions to the corresponding linear equation, we define a set of time-weighted Sobolev spaces. Under small condition on the initial value, we prove the global existence and asymptotic behavior of the solution in the corresponding Sobolev spaces by the contraction mapping principle.     

The behavior of the best Sobolev trace constant and extremals in thin domains

In this paper, we study the asymptotic behavior of the best Sobolev trace constant and extremals for the immersion W1,p(Ω)?Lq(∂Ω) in a bounded smooth domain when it is contracted in one direction. We find that the limit problem, when rescaled in a suitable way, is a Sobolev-type immersion in weighted spaces over a projection of Ω, W1,p(P(Ω),α)?Lq(P(Ω),β).For the special case p=q, this problem leads to an eigenvalue problem with a nonlinear boundary condition. We also study the convergence of the eigenvalues and eigenvectors in this case.     

Domain perturbations and estimates for the solutions of second order elliptic equations

We study the dependence of the variational solution of the inhomogeneous Dirichlet problem for a second order elliptic equation with respect to perturbations of the domain. We prove optimal L² and energy estimates for the difference of two solutions in two open sets in terms of the “distance” between them and suitable geometrical parameters which are related to the regularity of their boundaries. We derive such estimates when at least one of the involved sets is uniformly Lipschitz: due to the connection of this problem with the regularity properties of the solutions in the L² family of Sobolev–Besov spaces, the Lipschitz class is the reasonably weakest one compatible with the optimal estimates.