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This work is devoted to study the relation between regularity and decay of solutions of some dissipative perturbations of the Korteweg-de Vries equation in weighted and asymmetrically weighted Sobolev spaces. [ABSTRACT FROM AUTHOR]

A class of fourth-order strictly pseudohyperbolic operators with variable coefficients is considered. Energy estimates are established under certain conditions on the coefficients. These estimates imply the uniqueness of the solution to the Cauchy problem and a priori estimates. [ABSTRACT FROM AUTHOR]

We study nonlocal elliptic and parabolic equations on C1,τ open sets in weighted Sobolev spaces, where τ ∈ (0, 1). The operators we consider are infinitesimal generators of symmetric stable Lévy processes, whose Lévy measures are allowed to be very singular. Additionally, for parabolic equations, the measures are assumed to be merely measurable in the time variable. [ABSTRACT FROM AUTHOR]

The novelty of this note is to establish existence result for the following anisotropic elliptic equation ... where the datum f ∈ L¹ (Ω) and ... and H (x,ϑ) ∈ L¹ (Ω). Furthermore only the large monotonicity conditions will be assumed on B(x; s; ξ). To overcome this difficulty we will use the approach of Minty's lemma in the anisotropic weighted Sobolev spaces. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the existence and uniqueness of weak solution for the problem...is bounded and open. Here, the functions f: O R and g: O × R R satisfy the suitable hypotheses. [ABSTRACT FROM AUTHOR]

This paper introduces a novel higher order Adams inequality that incorporates an exact growth condition for a class of weighted Sobolev spaces. Our rigorous proof confirms the validity of this inequality and provides insights into the optimal nature of the critical constant and the exponent within the denominator. Furthermore, we apply this inequality to study a class of ordinary differential equations (ODEs), where we successfully derive both a concept of the weak solution and a comprehensive regularity theory. [ABSTRACT FROM AUTHOR]

In an open, bounded Lipschitz polygon Ω ⊂ R 2 , we establish weighted analytic regularity for a semilinear, elliptic PDE with analytic nonlinearity and subject to a source term f which is analytic in Ω . The boundary conditions on each edge of ∂ Ω are either homogeneous Dirichlet or homogeneous Neumann BCs. The presently established weighted analytic regularity of solutions implies exponential convergence of various approximation schemes: hp-finite elements, reduced order models via Kolmogorov n-widths of solution sets in H 1 (Ω) , quantized tensor formats and certain deep neural networks. [ABSTRACT FROM AUTHOR]

In this paper, we use a weighted version of Poincaré's inequality to study density and extension properties of weighted Sobolev spaces over some open set Ω⊆RN$\Omega \subseteq \mathbb {R}^N$. Additionally, we study the specific case of monomial weights w(x1,...,xN)=∏i=1Nxiai,ai≥0$w(x_1,\ldots ,x_N)=\prod _{i=1}^N\left|x_i \right|^{a_i},\ a_i\ge 0$, showing the validity of a weighted Poincaré inequality together with some embedding properties of the associated weighed Sobolev spaces. [ABSTRACT FROM AUTHOR]

The retrospective inverse problem for evolution equations is formulated as the reconstruction of unknown initial data by a given solution at the final time. We consider the inverse retrospective problem for a one-dimensional parabolic equation in two disconnected intervals with weak solutions in weighted Sobolev spaces. The two solutions are connected with nonstandard interface conditions, and thus this problem is solved in the whole spatial region. Such a problem, as with other inverse problems, is ill-posed, and for its numerical solution, specific techniques have to be used. The direct problem is first discretized by a difference scheme which provides a second order of approximation in space. For the resulting ordinary differential equation system, the positive coerciveness is established. Next, we develop an iterative conjugate gradient method to solve the ill-posed systems of the difference equations, which are obtained after weighted time discretization, of the inverse problem. Test examples with noisy input data are discussed. [ABSTRACT FROM AUTHOR]

We consider a class of strictly pseudohyperbolic equations with lower order terms. The solvability of the Cauchy problem in Sobolev spaces with special weights is established. The uniqueness of the solution is proven and estimates are obtained. [ABSTRACT FROM AUTHOR]

We study a class of second-order degenerate linear parabolic equations in divergence form in (-\infty, T) \times {\mathbb {R}}^d_+ with homogeneous Dirichlet boundary condition on (-\infty, T) \times \partial {\mathbb {R}}^d_+, where {\mathbb {R}}^d_+ = \{x \in {\mathbb {R}}^d: x_d>0\} and T\in {(-\infty, \infty ]} is given. The coefficient matrices of the equations are the product of \mu (x_d) and bounded uniformly elliptic matrices, where \mu (x_d) behaves like x_d^\alpha for some given \alpha \in (0,2), which are degenerate on the boundary \{x_d=0\} of the domain. Our main motivation comes from the analysis of degenerate viscous Hamilton-Jacobi equations. Under a partially VMO assumption on the coefficients, we obtain the well-posedness and regularity of solutions in weighted Sobolev spaces. Our results can be readily extended to systems. [ABSTRACT FROM AUTHOR]

We study the sharp constant for the embedding of W^{1,p}_0(\Omega) into L^q(\Omega), in the case 2p and q is sufficiently close to p, extremal functions attaining the sharp constant are unique, up to a multiplicative constant. This in turn gives the uniqueness of solutions with minimal energy to the Lane-Emden equation, with super-homogeneous right-hand side. The result is achieved by suitably adapting a linearization argument due to C.-S. Lin. We rely on some fine estimates for solutions of p-Laplace–type equations by L. Damascelli and B. Sciunzi. [ABSTRACT FROM AUTHOR]

We study the $\Gamma$ -convergence of nonconvex vectorial integral functionals whose integrands satisfy possibly degenerate growth and coercivity conditions. The latter involve suitable scale-dependent weight functions. We prove that under appropriate uniform integrability conditions on the weight functions, which shall belong to a Muckenhoupt class, the corresponding functionals $\Gamma$ -converge, up to subsequences, to a degenerate integral functional defined on a limit weighted Sobolev space. The general analysis is then applied to the case of random stationary integrands and weights to prove a stochastic homogenization result for the corresponding functionals. [ABSTRACT FROM AUTHOR]

We present a highly accurate and efficient spectral Galerkin method for the advection–diffusion–reaction equations with fractional lower-order terms in one dimension. We first prove sharp a priori regularity estimates of solutions in weighted Sobolev spaces. Then we obtain optimal convergence orders of the spectral Galerkin method. We also describe its implementation and present an efficient solver with a quasi-optimal complexity. Finally, we perform the numerical experiments and report numerical observations to support the theoretical predictions. [ABSTRACT FROM AUTHOR]

We prove the existence and uniqueness of a weak solution to a Dirichlet boundary value problem for a class of nonlinear degenerate elliptic equations in the setting of weighted Sobolev spaces. Our proof is based on the weighted Sobolev spaces theory and the Browder–Minty theorem. First, we transform the problem into an equivalent operator equation; second, we utilized the Browder–Minty theorem to prove the existence and uniqueness of a weak solution to the considered problem. [ABSTRACT FROM AUTHOR]

We study the homogenization of the Poisson equation with a reaction term and of the eigenvalue problem associated to the generator of multiscale Langevin dynamics. Our analysis extends the theory of two-scale convergence to the case of weighted Sobolev spaces in unbounded domains. We provide convergence results for the solution of the multiscale problems above to their homogenized surrogate. A series of numerical examples corroborate our analysis. [ABSTRACT FROM AUTHOR]

In this work, we study the very weak solution for the stationary exterior Stokes problem with Navier slip boundary condition. We try to investigate some results of the existence and the uniqueness of solutions related to this problem in an exterior domain in Lp$$ {L}^p $$‐theory where 1

In this work, we prove the existence of a weak solutions for the initial boundary value problem associated with the nonlinear degenerate parabolic equations ∂u... div b(x, t, u, Vu) = φ(x, t) + div a(x, t, Vu). ∂t We will use the Topological degree theory for operators of the type T + S, where S is a bounded demicontinuous map of type (S+) and T is a linear densely defined maximal monotone map with respect to a domain of T, and we study this problem in the space Lp(0, T; Wj,p(Ω, ω)), where Ω is a bounded domain in RN (N ≥ 2), p≥2and ω isavectorofweightfunctions. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence and multiplicity results for the semilinear subelliptic equation with free perturbation term. By using the degenerate Rellich-Kondrachov compact embedding theorem, the precise lower bound estimate of Dirichlet eigenvalues for the finitely degenerate elliptic operator and minimax method, we obtain the existence and multiplicity of weak solutions for the problem. [ABSTRACT FROM AUTHOR]

In this work we consider equations of the form ∂tu+P(∂x)u+G(u,∂xu,⋯,∂xlu)=0,$$\begin{equation*}\hskip7pc \partial _t u+P\big (\partial _x\big) u+G\big (u,\partial _xu,\dots ,\partial _x^l u\big)=0, \end{equation*}$$where P is any polynomial without constant term, and G is any polynomial without constant or linear terms. We prove that if u is a sufficiently smooth solution of the equation, such that suppu(0),suppu(T)⊂(−∞,B]$\operatorname{supp}u(0),\operatorname{supp}u(T)\subset { (-\infty ,B ]}$ for some B>0$B>0$, then there exists R0>0$R_0>0$ such that suppu(t)⊂(−∞,R0]$\operatorname{supp}u(t)\subset (-\infty ,R_0]$ for every t∈[0,T]$t\in [0,T]$. Then, as an example of the application of this result, we employ it to show a unique continuation principle for the Kawahara equation, ∂tu+∂x5u+∂x3u+u∂xu=0,$$\begin{equation*}\hskip9pc \partial _t u+\partial _x^5 u+\partial _x^3 u+u\partial _x u=0, \end{equation*}$$and for the generalized KdV hierarchy ∂tu+(−1)k+1∂x2k+1u+G(u,∂xu,⋯,∂x2ku)=0.$$\begin{equation*}\hskip6pc \partial _t u+ (-1)^{k+1}\partial _x^{2k+1} u+G\big (u,\partial _x u,\dots , \partial _x^{2k}u\big) =0. \end{equation*}$$ [ABSTRACT FROM AUTHOR]

We deal with the weak sequential density of smooth maps in the fractional Sobolev classes of W s , p maps in high dimension domains and with values into the circle. When s is lower than one, using interpolation theory we introduce a natural energy in terms of optimal extensions on suitable weighted Sobolev spaces. The relaxation problem is then discussed in terms of Cartesian currents. When s p = 1 , the energy gap of the relaxed functional is always finite and is given by the minimal connection of the singularities times an energy weight, obtained through a minimum problem for one dimensional W 1 / p , p maps with degree one. When s p > 1 , instead, concentration on codimension one sets needs unbounded energy. We finally treat the case where s is greater than one, obtaining an almost complete picture. [ABSTRACT FROM AUTHOR]

The retrospective inverse problem for evolution equations is formulated as the reconstruction of unknown initial data by a given solution at the final time. We consider the inverse retrospective problem for a one-dimensional parabolic equation in two disconnected intervals with weak solutions in weighted Sobolev spaces. The two solutions are connected with nonstandard interface conditions, and thus this problem is solved in the whole spatial region. Such a problem, as with other inverse problems, is ill-posed, and for its numerical solution, specific techniques have to be used. The direct problem is first discretized by a difference scheme which provides a second order of approximation in space. For the resulting ordinary differential equation system, the positive coerciveness is established. Next, we develop an iterative conjugate gradient method to solve the ill-posed systems of the difference equations, which are obtained after weighted time discretization, of the inverse problem. Test examples with noisy input data are discussed.

We consider a class of strictly pseudohyperbolic equations with lower order terms. The solvability of the Cauchy problem in Sobolev spaces with special weights is established. The uniqueness of the solution is proven and estimates are obtained.

In this paper we study the problem of the best approximation by linear methods of solutions to one Triebel-type equation. This problem was solved by using estimates of the linear widths of the unit ball in corresponding spaces of differentiable functions. According to the definition, linear widths give the best estimates for the approximation of compact sets in a given normed space by linear methods which are implemented through finite-dimensional operators. The problem includes answers to the questions about the solvability of the studied equation, the construction of the corresponding weighted space of differentiable functions, the development of a method for estimating linear widths of compact sets in weighted polynomial Sobolev space. In this work, conditions are obtained under which the considered operator has a bounded inverse. The weighted Sobolev space corresponding to the posed problem is determined. Upper estimates are obtained for the counting function for a sequence of linear widths, which correspond to the posed problem. One example is constructed in which two-sided estimates of linear widths are given. The method for solving this problem can be applied to the numerical solution of non-standard ordinary differential equations on an infinite axis.

We prove weighted analytic regularity of solutions to the Dirichlet problem for the integral fractional Laplacian in polygons with analytic right-hand side. We localize the problem through the Caffarelli-Silvestre extension and study the tangential differentiability of the extended solutions, followed by bootstrapping based on Caccioppoli inequalities on dyadic decompositions of vertex, edge, and vertex-edge neighborhoods. [ABSTRACT FROM AUTHOR]

We provide an extension operator for weighted Sobolev spaces on polyhedral cones K involving a mixture of weights, which measure the distance to the vertex and the edges of the cone, respectively. Our results are based on Stein's extension operator for Sobolev spaces and generalize previous results of the first author. [ABSTRACT FROM AUTHOR]

We study the regularity of solutions of the Poisson equation with Dirichlet, Neumann and mixed boundary values in polyhedral cones K ⊂ ℝ 3 in the specific scale   B τ , τ α ,   1 τ = α d + 1 p   of Besov spaces. The regularity of the solution in these spaces determines theorder of approximation that can be achieved by adaptive and nonlinear numerical schemes. We aim for a thorough discussion of homogeneous and inhomogeneous boundary data in all settings studied and show that the solutions are much smoother in this specific Besov scale compared to the fractional Sobolev scale H s in all cases,which justifies the use of adaptive schemes. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the existence of solutions for a coupled system of integral equations in the Cartesian product of weighted Sobolev spaces ε = Wω1,1 (a, b). The results were achieved by equipping the space E with the vector-valued norms and using the measure of noncompactness constructed in [F.P. Najafabad, J.J. Nieto, H.A. Kayvanloo, Measure of noncompactness on weighted Sobolev space with an application to some nonlinear convolution type integral equations, J. Fixed Point Theory Appl., 22(3), 75, 2020] to applicate the generalized Darbo's fixed point theorem [J.R. Graef, J. Henderson, and A. Ouahab, Topological Methods for Differential Equations and Inclusions, CRC Press, Boca Raton, FL, 2018]. [ABSTRACT FROM AUTHOR]

We present a weighted L p L p -theory of parabolic systems on a half space R d + R + d. The leading coefficients are assumed to be only measurable in time t t and have small bounded mean oscillations (BMO) with respect to the spatial variables x x , and the lower order coefficients are allowed to blow up near the boundary. [ABSTRACT FROM AUTHOR]

In this paper, we consider the stationary Stokes equations in an exterior domain three-dimensional under a slip boundary condition without friction. We set the problem in weighted Sobolev spaces in order to control the behavior at infinity of the solutions. In this work, we try to investigate the existence and uniqueness of the weak solutions related to this problem in L p - theory when p > 3 . Our proof is based on obtaining Inf-Sup conditions that play a fundamental role. [ABSTRACT FROM AUTHOR]

We study covariant Sobolev spaces and ∇ -differential operators with coefficients in general Hermitian vector bundles on Riemannian manifolds, stressing a coordinate-free approach that uses connections (which are typically denoted ∇ ). These concepts arise naturally from geometric partial differential equations, including some that are formulated on plain Euclidean domains, for instance, from problems formulated on the boundary of smooth domains or in relation to the weighted Sobolev spaces used to study PDEs on polyhedral domains. We prove several basic properties of the covariant Sobolev spaces and of the ∇ -differential operators on general manifolds. For instance, we prove mapping properties for our differential operators and the independence of the covariant Sobolev spaces on the choices of the connection ∇ , as long as the new connection is obtained using a totally bounded perturbation. We also introduce the Fréchet finiteness condition (FFC) for totally bounded vector fields, which is satisfied, for instance, by open subsets of manifolds with bounded geometry. When (FFC) is satisfied, we provide several equivalent definitions of our covariant Sobolev spaces and of our ∇ -differential operators. We also introduce and study the notion of a ∇ -bidifferential operator (a bilinear version of differential operators), obtaining results similar to those obtained for ∇ -differential operators. Bilinear differential operators are necessary for a global, geometric discussion of variational problems. We tried to write the paper so that it is accessible to a large audience. [ABSTRACT FROM AUTHOR]

We show that the Benjamin equation is globally well-posed for real-valued data in the weighted space H s ∩ H r s − 2 r ≔ { u | ‖ u ‖ H s (R x) + ‖ u ˆ ‖ H r (R ξ + , (1 + | ξ |) 2 (s − 2 r) d ξ) < ∞ } , where 0 ≤ r and − 3 4 + r < s. The proof is based on direct extensions of standard linear and bilinear estimates originated in Kenig et al. (1993), Kenig et al. (1996), Linares (1999), Kozono et al. (2001), Colliander et al. (2003), Li and Wu (2010) to the weighted settings. [ABSTRACT FROM AUTHOR]

We consider the initial value problem associated to the low dispersion fractionary Benjamin–Bona–Mahony equation, fBBM. Our aim is to establish local persistence results in weighted Sobolev spaces and to obtain unique continuation results that imply that those results above are sharp. Hence, arbitrary polynomial type decay is not preserved by the fBBM flow. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the existence of solutions for a coupled system of integral equations in the Cartesian product of weighted Sobolev spaces E = Wω1,1 (a,b) x Wω1,1 (a,b). The results were achieved by equipping the space E with the vector-valued norms and using the measure of noncompactness constructed in [F.P. Najafabad, J.J. Nieto, H.A. Kayvanloo, Measure of noncompactness on weighted Sobolev space with an application to some nonlinear convolution type integral equations, J. Fixed Point Theory Appl., 22(3), 75, 2020] to applicate the generalized Darbo’s fixed point theorem [J.R. Graef, J. Henderson, and A. Ouahab, Topological Methods for Differential Equations and Inclusions, CRC Press, Boca Raton, FL, 2018].

This study typically emphasizes analyzing the geometrical singularities of weak solutions of the mixed boundary value problem for the stationary Stokes and Navier-Stokes system in two-dimensional non-smooth domains with corner points and points at which the type of boundary conditions change. The existence of these points on the boundary generally generates local singularities in the solution. We will see the impact of the geometrical singularities of the boundary or the mixed boundary conditions on the qualitative properties of the solution including its regularity. The solvability of the underlying boundary value problem is analyzed in weighted Sobolev spaces and the regularity theorems are formulated in the context of these spaces. To compute the singular terms for various boundary conditions, the generalized form of the boundary eigenvalue problem for the stationary Stokes system is derived. The emerging eigenvalues and eigenfunctions produce singular terms, which permits us to evaluate the optimal regularity of the corresponding weak solution of the Stokes system. Additionally, the obtained results for the Stokes system are further extended for the non-linear Navier-Stokes system.

The main purpose of this paper is to investigate the Oseen problem in R 2. The established results are related to the existence and the uniqueness of weak, strong and very weak solutions on L p -theory for any real 1 < p < ∞. Since the domain is considered as unbounded, we set the problem in weighted spaces in order to control the behaviors at infinity of functions. [ABSTRACT FROM AUTHOR]