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In this paper we are interested in the existence of solutions for Navier problem associated with the degenerate nonlinear elliptic equations in the setting of the weighted Sobolev spaces.

We consider a system of $N$ neurons, each spiking randomly with rate depending on its membrane potential. When a neuron spikes, its potential is reset to $0$ and all other neurons receive an additional amount $h/N$ of potential, where $ h > 0$ is some fixed parameter. In between successive spikes, each neuron's potential undergoes some leakage at constant rate $ \alpha. $ While the propagation of chaos of the system, as $N \to \infty$, to a limit nonlinear jumping stochastic differential equation has already been established in a series of papers, see De Masi et al. (2015) and Fournier and L\"ocherbach (2016), the present paper is devoted to the associated central limit theorem. More precisely we study the measure valued process of fluctuations at scale $ N^{-1/2}$ of the empirical measures of the membrane potentials, centered around the associated limit. We show that this fluctuation process, interpreted as c\`adl\`ag process taking values in a suitable weighted Sobolev space, converges in law to a limit process characterized by a system of stochastic differential equations driven by Gaussian white noise. We complete this picture by studying the fluctuations, at scale $ N^{-1/2}, $ of a fixed number of membrane potential processes around their associated limit quantities, giving rise to a mesoscopic approximation of the membrane potentials that take into account the correlations within the finite system.

When solving elliptic partial differential equations in a region containing immersed interfaces (possibly evolving in time), it is often desirable to approximate the problem using an independent background discretisation, not aligned with the interface itself. Optimal convergence rates are possible if the discretisation scheme is enriched by allowing the discrete solution to have jumps aligned with the surface, at the cost of a higher complexity in the implementation. A much simpler way to reformulate immersed interface problems consists in replacing the interface by a singular force field that produces the desired interface conditions, as done in immersed boundary methods. These methods are known to have inferior convergence properties, depending on the global regularity of the solution across the interface, when compared to enriched methods. In this work we prove that this detrimental effect on the convergence properties of the approximate solution is only a local phenomenon, restricted to a small neighbourhood of the interface. In particular we show that optimal approximations can be constructed in a natural and inexpensive way, simply by reformulating the problem in a distributionally consistent way, and by resorting to weighted norms when computing the global error of the approximation., Comment: 27 pages, 10 figures, 4 tables

We study a nonlinear Schrodinger–Poisson system which reduces to the nonlinear and nonlocal PDE $$\begin{aligned} - \Delta u+ u + \lambda ^2 \left( \frac{1}{\omega |x|^{N-2}}\star \rho u^2\right) \rho (x) u = |u|^{q-1} u \quad x \in {{\mathbb {R}}}^N, \end{aligned}$$ where $$\omega = (N-2)|{\mathbb {S}}^{N-1} |,$$ $$\lambda >0,$$ $$q\in (1,2^{*} -1),$$ $$\rho :{{\mathbb {R}}}^N \rightarrow {{\mathbb {R}}}$$ is nonnegative, locally bounded, and possibly non-radial, $$N=3,4,5$$ and $$2^*=2N/(N-2)$$ is the critical Sobolev exponent. In our setting $$\rho $$ is allowed as particular scenarios, to either (1) vanish on a region and be finite at infinity, or (2) be large at infinity. We find least energy solutions in both cases, studying the vanishing case by means of a priori integral bounds on the Palais–Smale sequences and highlighting the role of certain positive universal constants for these bounds to hold. Within the Ljusternik–Schnirelman theory we show the existence of infinitely many distinct pairs of high energy solutions, having a min–max characterisation given by means of the Krasnoselskii genus. Our results cover a range of cases where major loss of compactness phenomena may occur, due to the possible unboundedness of the Palais–Smale sequences, and to the action of the group of translations.

Given three measurable functions V r ≥ 0 , K r > 0 and Q r ≥ 0 , r > 0 , we consider the bilaplacian equation Δ 2 u + V ( | x | ) u = K ( | x | ) f ( u ) + Q ( | x | ) in R N and we find radial solutions thanks to compact embeddings of radial spaces of Sobolev functions into sum of weighted Lebesgue spaces.

The Purkinje network is the specialized conduction system in the heart. It ensures the physiological spread of the electrical wave in the ventricles. In this work, in an insulated heart framework, we model the free running Purkinje system, using the monodomain equation. The intra-myocardium part of the Purkinje fiber is coupled to the ventricular tissue using the bidomain equation. The coupling is performed through the extracellular potential. We discretize the problem in time using a semi-implicit scheme. Then, we write a variational formulation of the semi discrete problem in a non standard weighted Sobolev functional spaces. We prove the existence and uniqueness of the solution of the Purkinje/myocardium semi-discretized problem. We discretize in space by the finite element P 1 − L a g r a n g e and conduct some numerical tests showing the anterograde and retrograde propagation of the electrical wave between the tissue and the Purkinje fibers.

The Cauchy problem for a quasi-linear parabolic equation with a small parameter multiplying a higher derivative is considered in two cases when the solution of the limit problem has a point of gradient catastrophe. Asymptotic solutions are found by using the Cole–Hopf transform. The integrals determining the asymptotic solutions correspond to the Lagrange singularities of type \(A\) and the boundary singularities of type \(B\). The behavior of the asymptotic solutions is described in terms of the weighted Sobolev spaces.

We study existence of solutions for a boundary degenerate (or singular) quasilinear equation in a smooth bounded domain under Dirichlet boundary conditions. We consider a weighted p -Laplacian operator with a coefficient that is locally bounded inside the domain and satisfying certain additional integrability assumptions. Our main result applies for boundary value problems involving continuous non-linearities having no growth restriction, but provided the existence of a sub and a supersolution is guaranteed. As an application, we present an existence result for a boundary value problem with a non-linearity f ( u ) satisfying f ( 0 ) ≤ 0 and having ( p − 1 ) -sublinear growth at infinity.

© 2021 The Authors. The aim of this paper is to develop a layer potential theory in L2-based weighted Sobolev spaces on Lipschitz bounded and exterior domains of Rn , n ≥ 3, for the anisotropic Stokes system with L∞ viscosity tensor coefficient satisfying an ellip- ticity condition for symmetric matrices with zero matrix trace. To do this, we explore equivalent mixed variational formulations and prove the well-posedness of some transmission problems for the anisotropic Stokes system in Lipschitz domains of Rn, with the given data in L2-based weighted Sobolev spaces. These results are used to define the volume (Newtonian) and layer potentials and to obtain their properties. Then, we analyze the well-posedness of the exterior Dirichlet and Neumann problems for the anisotropic Stokes system with L∞ symmetrically elliptic tensor coefficient by representing their solutions in terms of the obtained volume and layer potentials. EPSRC grant EP/M013545/1: "Mathematical Analysis of Boundary-Domain Integral Equations for Nonlinear PDEs"; Babeş-Bolyai University research grant AGC35124/31.10.2018; Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy-EXC 2075-390740016.

We study a Stokes flow in a cylindrical-type fractal domain with homogeneous Dirichlet boundary conditions. We consider its numerical approximation by mixed methods: finite elements in space and finite differences in time. We introduce a suitably refined mesh a la Grisvard, which in turn will allow us to obtain an optimal a priori error estimate.

We consider the initial value problem for a class of semilinear p-evolution equations with (t, x)-depending coefficients. Under suitable decay conditions for |x|→∞ on the imaginary part of the coefficients, we prove local in time well posedness of the Cauchy problem in suitable weighted Sobolev spaces.

We consider the spatial approximation of the Cauchy problem for a linear uniformly parabolic PDE of second order, with nondivergent operator and unbounded time- and space-dependent coefficients, where equation’s free term and initial data are also allowed to grow. We concentrate on the special case where the PDE has one dimension in space. As in [10], we consider a suitable variational framework and approximate the PDE problem’s generalised solution in the spatial variable, with the use of finite-difference methods, but we obtain, for this case, consistency and convergence results sharper than the corresponding results obtained in [10] for the more general multidimensional case. info:eu-repo/semantics/publishedVersion

We study the properties of the weighted space Hk 2α(Ω) and weighted set Wk 2α(Ω, δ)for boundary value problem with singularity.

In this paper, we prove the existence of radial solutions for the nonlinear elliptic problem - div ⁢ ( A ⁢ ( | x | ) ⁢ ∇ ⁡ u ) + V ⁢ ( | x | ) ⁢ u = K ⁢ ( | x | ) ⁢ f ⁢ ( u ) in ⁢ ℝ N , -\mathrm{div}(A(\lvert x\rvert)\nabla u)+V(\lvert x\rvert)u=K(\lvert x\rvert)f% (u)\quad\text{in }\mathbb{R}^{N}, with suitable hypotheses on the radial potentials A, V, K. We first get compact embeddings of radial weighted Sobolev spaces into sums of weighted Lebesgue spaces, and then we apply standard variational techniques to get existence results.

In this work, we will prove the existence of bounded solutions for the nonlinear elliptic equations - div(a(x,u,\nabla u)) = g(x,u,\nabla u) -divf,−div(a(x,u,∇u))=g(x,u,∇u)−divf, in the setting of the weighted Sobolev space W^{1,p}(\Omega,w)W 1,p (Ω,w) where aa, gg are Caratheodory functions which satisfy some conditions and ff satisfies suitable summability assumption.

In this work we are interested in the existence and uniqueness of solutions for the Navier problem associated to the degenerate nonlinear elliptic equations Δ (v ( x ) | Δ u | r − 2 Δ u ) − ∑ j = 1 n D j [ w 1 ( x ) 𝒜 j ( x , u , ∇ u ) ] + b ( x , u , ∇ u ) w 2 ( x ) = f 0 ( x ) − ∑ j = 1 n D j f j ( x ) , in Ω $$\matrix{{\Delta {\rm{(v}}({\rm{x}})\left| {\Delta {\rm{u}}} \right|^{{\rm{r}} - 2} \Delta {\rm{u}}) - \sum\limits_{{\rm{j}} = 1}^{\rm{n}} {{\rm{D}}_{\rm{j}} [{\rm{w}}_1 ({\rm{x}}){\cal{A}}_{\rm{j}} ({\rm{x}},{\rm{u}},\nabla {\rm{u}})]} } \hfill \cr { + \;{\rm{b}}({\rm{x}},{\rm{u}},\nabla {\rm{u}})\;{\rm{w}}_2 ({\rm{x}}) = {\rm{f}}_0 ({\rm{x}}) - \sum\limits_{{\rm{j}} = 1}^{\rm{n}} {{\rm{D}}_{\rm{j}} {\rm{f}}_{\rm{j}} ({\rm{x}}),\;\;\;\;\;{\rm{in}}\;\Omega } }}$$ in the setting of the Weighted Sobolev Spaces.

The classical assumption in sampling and spline theories is that the input signal is square-integrable, which prevents us from applying such techniques to signals that do not decay or even grow at infinity. In this paper, we develop a sampling theory for multidimensional non-decaying signals living in weighted $ L _{ p } $ spaces. The sampling and reconstruction of an analog signal can be done by a projection onto a shift-invariant subspace generated by an interpolating kernel. We show that, if this kernel and its biorthogonal counterpart are elements of appropriate hybrid-norm spaces, then both the sampling and the reconstruction are stable. This is an extension of earlier results by Aldroubi and Gröchenig. The extension is required because it allows us to develop the theory for the ideal sampling of non-decaying signals in weighted Sobolev spaces. When the d-dimensional signal and its d∕p + ε derivatives, for arbitrarily small ε > 0, grow no faster than a polynomial in the $ L _{ p } $ sense, the sampling operator is shown to be bounded even without a sampling kernel. As a consequence, the signal can also be interpolated from its samples with a nicely behaved interpolating kernel.

International audience; We give short simple proofs of Uspenskii's results characterizing Besov spaces as trace spaces of weighted Sobolev spaces. We generalize Uspenskii's results and prove the optimality of these generalizations. We next show how classical results on the functional calculus in the Besov spaces can be obtained as straightforward consequences of the theory of weighted Sobolev spaces.

In our recent work, the sampling and reconstruction of non-decaying signals, modeled as members of weighted- $$L_p$$ spaces, were shown to be stable with an appropriate choice of the generating kernel for the shift-invariant reconstruction space. In this paper, we extend the Strang–Fix theory to show that, for d-dimensional signals whose derivatives up to order L are all in some weighted- $$L_p$$ space, the weighted norm of the approximation error can be made to go down as $$O(h^L)$$ when the sampling step h tends to 0. The sufficient condition for this decay rate is that the generating kernel belongs to a particular hybrid-norm space and satisfies the Strang–Fix conditions of order L. We show that the $$O(h^L)$$ behavior of the error is attainable for both approximation schemes using projection (when the signal is prefiltered with the dual kernel) and interpolation (when a prefilter is unavailable). The requirement on the signal for the interpolation method, however, is slightly more stringent than that of the projection because we need to increase the smoothness of the signal by a margin of $$d/p+\varepsilon $$ , for arbitrary $$\varepsilon >0$$ . This extra amount of derivatives is used to make sure that the direct sampling is stable.

We establish uniform estimates in weighted Sobolev spaces for the solutions of the Dirichlet problems on snowflake pre-fractal domains.

The aim of this work is to study the optimal control problem associated to a linear parabolic equation with homogeneous Dirichlet boundary condition. The control variable is the matrix of \(L^1\)-coefficients in the main part of the parabolic operator. The precise answer about existence or none-existence of an \(L^1\)-optimal solution heavily depends on the class of admissible controls. The main questions concern the right setting of the optimal control problem with \(L^1\)-controls in the coefficients, and the right class of admissible solutions to the above problem. Using the direct method in the Calculus of variations, we discuss the solvability of the above optimal control problem in the class of \(H\)-admissible solutions.

We show that, under adequate norms, the Fourier transform is an isometry over a chain of nested weighted Sobolev spaces. As a result, an infinite number of useful Plancherel-like identities are derived. Possible extensions are discussed, giving rise to some open questions. (C) 2012 Elsevier Masson SAS. All rights reserved.

We study so{\`u}e infinite-horizon optimization problems on spaces of periodic functions for non periodic Lagrangians. The main strategy relies on the reduction to finite horizon thanks in the introduction of an avering operator.We then provide existence results and necessary optimality conditions in which the corresponding averaged Lagrangian appears.

We prove that weakly differentiable weights $w$ which, together with their reciprocals, satisfy certain local integrability conditions, admit a unique associated first-order $p$-Sobolev space, that is \[H^{1,p}(\mathbb{R}^d,w\,\d x)=V^{1,p}(\mathbb{R}^d,w\,\d x)=W^{1,p}(\mathbb{R}^d,w\,\d x),\] where $d\in\N$ and $p\in [1,\infty)$. If $w$ admits a (weak) logarithmic gradient $\nabla w/w$ which is in $L^q_{\text{loc}}(w\,\d x;\R^d)$, $q=p/(p-1)$, we propose an alternative definition of the weighted $p$-Sobolev space based on an integration by parts formula involving $\nabla w/w$. We prove that weights of the form $\exp(-\beta |\cdot|^q-W-V)$ are $p$-admissible, in particular, satisfy a Poincar\'e inequality, where $\beta\in (0,\infty)$, $W$, $V$ are convex and bounded below such that $|\nabla W|$ satisfies a growth condition (depending on $\beta$ and $q$) and $V$ is bounded. We apply the uniqueness result to weights of this type. The associated nonlinear degenerate evolution equation is also discussed., Comment: 23 pp., to appear in J. Funct. Anal. (in press)

The paper deals with the existence of entire solutions for a quasilinear equation \({(\mathcal E)_\lambda}\) in \({\mathbb{R}^N}\) , depending on a real parameter λ, which involves a general elliptic operator in divergence form A and two main nonlinearities. The competing nonlinear terms combine each other, being the first subcritical and the latter supercritical. We prove the existence of a critical value λ* > 0 with the property that \({(\mathcal E)_\lambda}\) admits nontrivial non-negative entire solutions if and only if λ ≥ λ*. Furthermore, when \({\lambda > \overline{\lambda} \ge \lambda^*}\) , the existence of a second independent nontrivial non-negative entire solution of \({(\mathcal{E})_\lambda}\) is proved under a further natural assumption on A.

This paper deals with entropy numbers and approximation numbers for compact embeddings of weighted Sobolev spaces into Lebesgue spaces in limiting situations. This work is based on related Hardy inequalities and the spectral theory of some degenerate elliptic operators.

In this paper we study a Dirichlet optimal control problem for a nonlinear monotone equation with degenerate weight function and with the coefficients which we adopt as controls in $L^\infty(\Omega)$. Since these types of equations can exhibit the Lavrentieff phenomenon, we consider the optimal control problem in coefficients in the so-called class of H-admissible solutions. Using the direct method of calculus of variations we discuss the solvability of the above optimal control problem, and prove the attainability of H-optimal pairs via optimal solutions of some nondegenerate perturbed optimal control problems. We also introduce the concept of the Mosco-stability for the above optimal control problem and study the variational properties of Mosco-stable problems with respect to the special type of domain perturbations.

In this paper, we consider the biharmonic problem of a partial differential inclusion with Dirichlet boundary conditions. We prove existence theorems for related partial differential inclusions with convex and nonconvex multivalued perturbations, and obtain an existence theorem on extremal solutions, and a strong relaxation theorem. Also we prove that the solution set is compact $R_{\delta}$ if the perturbation term of the related partial differential inclusion is convex, and its solution set is path-connected if the perturbation term is nonconvex.

We are concerned with the existence and non-existence of nontrivial weak solutions for a class of quasilinear scalar field equations in R N driven by competing nonlinearities with general potentials which can be unbounded or decaying to zero as | x | → + ∞ . Furthermore, the existence of ground states and/or bound states is considered.

Given α > 0 and f ∈ L 2 ( 0 , 1 ) , we are interested in the equation { − ( x 2 α u ′ ( x ) ) ′ + u ( x ) = f ( x ) on ( 0 , 1 ] , u ( 1 ) = 0 . We prescribe appropriate (weighted) homogeneous boundary conditions at the origin and prove the existence and uniqueness of H loc 2 ( 0 , 1 ] solutions. We study the regularity at the origin of such solutions. We perform a spectral analysis of the differential operator L u : = − ( x 2 α u ′ ) ′ + u under those appropriate homogeneous boundary conditions.

Global existence of weak solutions to the Navier–Stokes equations in a cylindrical domain under boundary slip conditions and with inflow and outflow is proved. To prove the energy estimate, crucial for the proof, we use the Hopf function. This makes it possible to derive an estimate such that the inflow and outflow need not vanish as t → ∞ . The proof requires estimates in weighted Sobolev spaces for solutions to the Poisson equation. Our result is the first step towards proving the existence of global regular special solutions to the Navier–Stokes equations with inflow and outflow.

In this work we are interested in the existence of solutions for Dirichlet problems associated with the degenerate semilinear elliptic equations − ∑ i , j = 1 n D j ( a i j ( x ) D i u ( x ) ) − μ u ( x ) g ( x ) = − f ( x , u ( x ) ) on Ω in the setting of the weighted Sobolev spaces W 0 1 , 2 ( Ω , ω ) .

The purpose of this work is to solve the exterior Stokes problem in the half-space \({\mathbb{R}{^n_+}}\) . We study the existence and the uniqueness of generalized solutions in weighted Lp theory with 1 < p < ∞. Moreover, we consider the case of strong solutions and very weak solutions. This paper extends the studies done in Alliot, Amrouche (Math. Methods Appl. 23:575–600, 2000) for an exterior Stokes problem in the whole space and in Amrouche, Bonzom (Exterior Problems in the Half-space, submitted) for the Laplace equation in the same geometry as here.

The purpose of this work is to solve exterior problems in the half-space for the Laplace operator. We give existence and unicity results in weighted L p 's theory with 1 p ∞ . This paper extends the studies done 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 (1) (1997) 55–81] with Dirichlet and Neumann conditions.

Let r=(x"1^2+x"2^2)^1^/^2 be the distance function to the origin [email protected]?R^2, and let us fix @d>0. We consider the ''Schrodinger-type mixed boundary value problem'' [email protected][email protected]^-^[email protected]?H^m^-^1(@W) on a bounded polygonal domain @[email protected]?R^2. The singularity in the potential @dr^-^2 severely limits the regularity of the solution u. This affects the rate of convergence to u of the finite element approximations u"[email protected]?S obtained using a quasi-uniform sequence of meshes. We show that a suitable graded sequence of meshes recovers the quasi-optimal convergence rate @?u-u"[email protected]?"H"^"1"("@W")@?Cdim(S"n)^-^m^/^[email protected][email protected]?"H"^"m"^"-"^"1"("@W"), where S"n are the FE spaces of continuous, piecewise polynomial functions of degree m>=1 associated to our sequence of meshes and u"n=u"S"""[email protected]?S"n are the FE approximate solutions. This is in spite of the fact that [email protected][email protected]?H^m^+^1(@W) in general. One of the main results of our paper is to show that the singularities due to the potential and the singularities due to the singularities of the domain or to the change in boundary conditions can be treated in the same way. Our proof is based on regularity and well-posedness results in weighted Sobolev spaces, with the weight taking into account all singularities (including the ones coming from the potential). Our regularity results apply also to operators with weaker singularities, like the Schrodinger operator [email protected][email protected]^-^1, for which we also obtain Fredholm conditions and a formula for the index. Our a priori estimates also extend to piecewise smooth domains (i.e., curvilinear polygonal domains).

The aim of this article is to study a nonlinear system modeling a Non-Newtonian fluid of polymer aqueous solutions. We are interested here in the existence of weak solutions for the stationary problem in a bounded plane domain or in two-dimensional exterior domain. Due to the third order of derivatives in the non-linear term, it's difficult to obtain solution satisfying energy inequality. But with a good choice of boundary conditions, an adapt special basis and the use of the good properties of the trilinear form associated to the non-linear term, we obtain energy solutions. The problem in bounded domain is treated and the more difficult problem on non bounded domain too., Comment: 20 pages

We deal with a weighted nonlinear Schrodinger-Poisson system, allowing the potentials to vanish at infinity.

Let Ω be a bounded domain in R n with C 2 and let u be a solution of the classical Poisson problem in ; i.e., { u = f in , u = 0 on , where f L p ( ) and is a weight in A p . The main goal of this paper is to prove the following a priori estimate u W 2 , p ( ) C f L p ( ), and to give some applications for weights given by powers of the distance to the boundary., Facultad de Ciencias Exactas

32 pages, no figures.-- MSC2000 codes: 41A10, 46E35, 46G10. MR#: MR2338656 (2008g:41004) Zbl#: Zbl pre05173014 We characterize the set of functions which can be approximated by continuous functions with the norm $\ \ {L infty(w)}$ for every weight w. This fact allows to determine the closure of the space of polynomials in $L infty(w)$ for every weight w with compact support. We characterize as well the set of functions which can be approximated by smooth functions with the norm $$ \ \ {W 1,\infty}(w_0,w_1)}\coloneq \ \ {L infty(w_0)}+ \ '\ {L infty(w_1)}, $$ for a wide range of (even non-bounded) weights $w_0,w_1$. We allow a great deal of independence among the weights. Research by first (A.P.), third (J.M.R.) and fourth (E.T.) autors was partially supported by three grants from MEC (MTM 2006-11976, MTM 2006-13000-C03-02, MTM 2006-26627-E), Spain. Publicado

We systematically study weighted Poincare type inequalities which are closely connected with Hardy type inequalities and establish the form of the optimal constants in some cases. Such inequalities are then used to relate entropy with entropy production and get intermediate asymptotics results for fast diffusion equations. To cite this article: A. Blanchet et al., C. R. Acad. Sci. Paris, Ser. I 344 (2007).

We are concerned with the boundary value problem for the steady Navier–Stokes equations in a 2D bounded domain with piecewise smooth boundary. Existence and uniqueness of the solution to the above problem is proved in weighted Sobolev spaces by means of the Mellin transform and the regularizer method.

We investigate the analytic regularity of the Stokes problem in a polygonal domain Ω⊂R2 with straight sides and piecewise analytic data. We establish a shift theorem in weighted Sobolev spaces of arbitrary order with explicit control of the order-dependence of the constants. The shift-theorem in the framework of countably weighted Sobolev spaces implies in particular interior analyticity and Gevrey-type analytic regularity near the corners.

Working in a weighted Sobolev space, a new result involving jumping nonlinearities for a semilinear elliptic boundary value problem in a bounded domain in R N is established. The nonlinear part of the equation is assumed to grow at most linearly and to be at resonance with the first eigenvalue of the linear part on the right. On the left, the nonlinearity crosses over (or jumps over) several higher eigenvalues. Existence is obtained through the use of infinite-dimensional critical point theory in the context of weighted Sobolev spaces and appears to be new even for the standard Dirichlet problem for the Laplacian.

In this work we present a homogenization result for a class of degenerate elliptic functionals mimicking strongly anisotropic media. We study the limit as ε→0 of the functionals [Formula: see text] where, for any ε>0, αε:ℝn×ℝn→ℝ, αε(x,ξ)≈ε(x)ξ,ξ>p/2-1, Aε∈Mn×n(ℝ) being measurable non-negative matrices such that [Formula: see text] almost everywhere, p>1. To take into account the anisotropy of the media we consider two families of weight functions reasonably different, λε and Λε, possibly degenerate or singular, such that: [Formula: see text] The convergence to the homogenized problem is obtained by a classical approach of Γ-convergence.

In this paper, we study existence of solutions to a class of semilinear degenerate elliptic equations in weighted Sobolev spaces.

Variations in harmonic amplitudes of one-dimensional recurrent signals of the types encountered in medical tonometry are analyzed over finite length records by single Fourier series. Two regularizational algorithms, one differentiation-invoking and the other differentiation-free, are advanced for the design of a universal amplitude spectral “tonometer” for such signals.

The density of polynomials is straightforward to prove in Sobolev spaces Wk,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 Lp spaces; besides, in the present paper we have proved some other results for weighted Sobolev spaces in infinite intervals.

RésuméSur une variété riemannienne complète de dimension n⩾3, on étudie le problème de la courbure scalaire prescrite, spécialement dans la cas nul. Sous certaines hypothèses, on montre, lorsque la courbure scalaire est nulle, l'existence de ε>0, tel que tout f∈C∞, vérifiant |f|0 et r est la distance à un point fixe.AbstractOn a complete Riemannian manifold of dimension n⩾3, we study the prescribed scalar curvature problem, especially in the null case. Under some hypotheses, we show, when the scalar curvature is zero, the existence of ε>0 such that any f∈C∞ satisfying |f|0 and r denotes the distance to a fixed point.

En este artículo, presentamos un estudio analítico de una clase de pesos, damos algunas propiedades y formulamos algunas estimaciones de estos. Definimos los correspondientes espacios de Sobolev con pesos, tratamos de establecer algunas relaciones entre los espacios de Sobolev con peso y sin peso. Construimos las inclusiones de Sobolev con peso entre los espacios de Sobolev con peso., In this paper, we present an analytical study of a class of weights, we formulate some properties and some estimates of these. From finimos corresponding Sobolev spaces with weights, try to establish some relations between Sobolev spaces with weight and weightless. We built Sobolev inclusions weighing between Sobolev spaces with weight.