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In this paper we study an optimal control problem for a linear boundary value problem with strongly degenerate coefficient in the main part of the elliptic operator and with the Neumann boundary control. Given a cost functional, the objective is to provide the well-posedness analysis of the corresponding optimal control problem, prove existence of the optimal solutions and propose the scheme for their approximation. [ABSTRACT FROM AUTHOR]

We show short-time existence for curves driven by curve diffusion flow with a prescribed contact angle α ∈ (0 , π) : The evolving curve has free boundary points, which are supported on a line and it satisfies a no-flux condition. The initial data are suitable curves of class W 2 γ with γ ∈ (3 2 , 2 ]. For the proof the evolving curve is represented by a height function over a reference curve: The local well-posedness of the resulting quasilinear, parabolic, fourth-order PDE for the height function is proven with the help of contraction mapping principle. Difficulties arise due to the low regularity of the initial curve. To this end, we have to establish suitable product estimates in time weighted anisotropic L 2 -Sobolev spaces of low regularity for proving that the non-linearities are well-defined and contractive for small times. [ABSTRACT FROM AUTHOR]

We obtain regularity results in weighted Sobolev spaces for the solution of the obstacle problem for the integral fractional Laplacian (− Δ) s in a Lipschitz bounded domain Ω ⊂ ℝ n satisfying the exterior ball condition. The weight is a power of the distance to the boundary ∂ Ω of Ω that accounts for the singular boundary behavior of the solution for any 0 < s < 1. These bounds then serve us as a guide in the design and analysis of a finite element scheme over graded meshes for any dimension n , which is optimal for n = 2. [ABSTRACT FROM AUTHOR]

Let X be a separable Banach space endowed with a nondegenerate centered Gaussian measure μ and let w be a positive function on X such that w ∈ W 1 , s (X , μ) and log w ∈ W 1 , t (X , μ) for some s > 1 and t > s ′ . In this paper, we introduce and study Sobolev spaces with respect to the weighted Gaussian measure ν : = w μ. We obtain results regarding the divergence operator (i.e. the adjoint in L 2 of the gradient operator along the Cameron–Martin space) and the trace of Sobolev functions on hypersurfaces x ∈ X | G (x) = 0 , where G is a suitable version of a Sobolev function. [ABSTRACT FROM AUTHOR]

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)|Δu|r−2Δu)−∑j=1nDj[w1(x)𝒜j(x,u,∇u)]+ b(x,u,∇u) w2(x)=f0(x)−∑j=1nDjfj(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.

In this work we present a newly developed study of the interpolation of weighted Sobolev spaces by the complex method. We show that in some cases, one can obtain an analogue of the famous Stein–Weiss theorem for weighted L p spaces. We consider an example which gives some indication that this may not be possible in all cases. Our results apply in cases which cannot be treated by methods in earlier papers about interpolation of weighted Sobolev spaces. They include, for example, a proof that [ W 1 , p (R d , ω 0) , W 1 , p (R d , ω 1) ] θ = W 1 , p (R d , ω 0 1 − θ ω 1 θ) whenever ω 0 and ω 1 are continuous and their quotient is the exponential of a Lipschitz function. We also mention some possible applications of such interpolation in the study of convergence in evolution equations. [ABSTRACT FROM AUTHOR]

We study a nonlocal Venttsel' problem in a nonconvex bounded domain with a Koch‐type boundary. Regularity results of the strict solution are proved in weighted Sobolev spaces. The numerical approximation of the problem is carried out, and optimal a priori error estimates are obtained. [ABSTRACT FROM AUTHOR]

We design and analyze several finite element methods (FEMs) applied to the Caffarelli–Silvestre extension that localizes the fractional powers of symmetric, coercive, linear elliptic operators in bounded domains with Dirichlet boundary conditions. We consider open, bounded, polytopal but not necessarily convex domains Ω ⊂ R d with d = 1 , 2 . For the solution to the Caffarelli–Silvestre extension, we establish analytic regularity with respect to the extended variable y ∈ (0 , ∞) . Specifically, the solution belongs to countably normed, power-exponentially weighted Bochner spaces of analytic functions with respect to y, taking values in corner-weighted Kondrat'ev-type Sobolev spaces in Ω . In Ω ⊂ R 2 , we discretize with continuous, piecewise linear, Lagrangian FEM ( P 1 -FEM) with mesh refinement near corners and prove that the first-order convergence rate is attained for compatible data f ∈ H 1 - s (Ω) with 0 < s < 1 denoting the fractional power. We also prove that tensorization of a P 1 -FEM in Ω with a suitable hp-FEM in the extended variable achieves log-linear complexity with respect to N Ω , the number of degrees of freedom in the domain Ω . In addition, we propose a novel, sparse tensor product FEM based on a multilevel P 1 -FEM in Ω and on a P 1 -FEM on radical-geometric meshes in the extended variable. We prove that this approach also achieves log-linear complexity with respect to N Ω . Finally, under the stronger assumption that the data be analytic in Ω ¯ , and without compatibility at ∂ Ω , we establish exponential rates of convergence ofhp-FEM for spectral fractional diffusion operators in energy norm. This is achieved by a combined tensor product hp-FEM for the Caffarelli–Silvestre extension in the truncated cylinder with anisotropic geometric meshes that are refined toward ∂ Ω . We also report numerical experiments for model problems which confirm the theoretical results. We indicate several extensions and generalizations of the proposed methods to other problem classes and to other boundary conditions on ∂ Ω . [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We study the regularity in weighted Sobolev spaces of Schrödinger-type eigenvalue problems, and we analyze their approximation via a discontinuous Galerkin (dG) h p finite element method. In particular, we show that, for a class of singular potentials, the eigenfunctions of the operator belong to analytic-type non-homogeneous weighted Sobolev spaces. Using this result, we prove that an isotropically graded h p dG method is spectrally accurate, and that the numerical approximation converges with exponential rate to the exact solution. Numerical tests in two and three dimensions confirm the theoretical results and provide an insight into the behavior of the method for varying discretization parameters. [ABSTRACT FROM AUTHOR]

We investigate a new hybrid method we call the discontinuous Galerkin‐inverted finite element method (DGIFEM) to approximate the solution of elliptic problems in Rn, especially when the growth or the decay of the solution is very slow. On the basis of both the discontinuous Galerkin discretization and the inverted finite element method, the DGIFEM keeps part of the domain bounded and maps the other infinite extent into a bounded region via a suitable polygonal inversion. The numerical solution is then constructed in an appropriate subspace of weighted Sobolev spaces, where the weights allow the control of the growth or the decay of functions at infinity. A careful study of the convergence of the DGIFEM is carried out and shows that the optimal order of convergence can always be reached. Finally, some numerical results are given as illustration of the good performance of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, a class of infinite horizon optimal control problems with a mixed control-state isoperimetrical constraint, also interpreted as a budget constraint, is considered. The underlying dynamics is assumed to be affine-linear in control. The crucial idea which is followed in this paper is the choice of a weighted Sobolev space as the state space. For this class of problems, we establish an existence result and apply it to a bilinear model of optimal cancer treatment with an isoperimetrical constraint including the overall amount of drugs used during the whole therapy horizon. A numerical analysis of this model is provided by means of open source software package OCMat, which implements a continuation method for solving discounted infinite horizon optimal control problems. [ABSTRACT FROM AUTHOR]

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 + ε , for arbitrary ε > 0 . This extra amount of derivatives is used to make sure that the direct sampling is stable. [ABSTRACT FROM AUTHOR]

Abstract Let L q , μ , 1 ≤ q ≤ ∞ , denote the weighted L q space of functions on the Euclidean unit ball B d with respect to the weight (1 − ‖ x ‖ 2 2) μ − 1 2 , μ ≥ 0 , and let W 2 , μ r be the weighted Sobolev space on B d equipped with a Gaussian measure ν. We investigate the probabilistic linear (n , δ) -widths λ n , δ (W 2 , μ r , ν , L q , μ) and the p -average linear n -widths λ n (a) (W 2 , μ r , ν , L q , μ) p , and obtain their asymptotic orders for all 1 ≤ q ≤ ∞ and 0 < p < ∞. [ABSTRACT FROM AUTHOR]

Abstract 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. [ABSTRACT FROM AUTHOR]

We consider the weighted parabolic problem of the type ut-div(ω2(x)|∇u|p-2∇u)=λω1(x)|u|p-2u,x∈Ω,u(x,0)=f(x),x∈Ω,u(x,t)=0,x∈∂Ω,t>0,for quite a general class of possibly unbounded weights ω1,ω2 satisfying the Hardy-type inequality. We prove existence of a global weak solution in the weighted Sobolev spaces provided that λ>0 is smaller than the optimal constant in the inequality. The domain is assumed to be bounded or quasibounded. The obtained solution is proven to belong to Lp(R+;W(ω1,ω2),01,p(Ω))∩L∞(R+;L2(Ω)). [ABSTRACT FROM AUTHOR]

We consider the Sobolev space over of square integrable functions whose gradient is also square integrable with respect to some positive weight. It is well known that smooth functions are dense in the weighted Sobolev space when the weight is uniformly bounded from below and above. This may not be the case when the weight is unbounded. In this paper, we focus on a class of two-dimensional weights where the density of smooth functions does not hold. This class was originally introduced by V.V. Zhikov; such weights have a unique singularity point of non-zero capacity. Following V.V. Zhikov, we first give a detailed analytical description of the weighted Sobolev space. Then we explain how to use Dirichlet forms theory to associate a diffusion process to such a degenerate non-regular space. [ABSTRACT FROM AUTHOR]

Abstract In this paper we investigate a mathematical model arising from volcanology describing surface deformation effects generated by a magma chamber embedded into Earth's interior and exerting on it a uniform hydrostatic pressure. The modeling assumptions translate mathematically into a Neumann boundary value problem for the classical Lamé system in a half-space with an embedded pressurized cavity. We establish well-posedness of the problem in suitable weighted Sobolev spaces and analyse the inverse problem of determining the pressurized cavity from partial measurements of the displacement field proving uniqueness and stability estimates. [ABSTRACT FROM AUTHOR]

In this work we study the initial-value problem for the fifth-order Korteweg-de Vries equation $$ \partial_{t}u+\partial_{x}^{5}u+u^k\partial_{x}u=0, \quad x,t\in \mathbb{R}, \; k=1,2, $$ in weighted Sobolev spaces $H^s(\mathbb{R})\cap L^2(\langle x \rangle^{2r}dx)$. We prove local and global results. For the case $k=2$ we point out the relationship between decay and regularity of solutions of the initial-value problem.

The goal of this work is to show that the generalized Korn inequality that replaces the symmetric part of the differential matrix in the classical Korn inequality by its trace‐free part is valid over John domains and weighted Sobolev spaces. The weights considered are nonnegative powers of the distance to the boundary. [ABSTRACT FROM AUTHOR]

We consider an evolution equation involving the fractional powers, of order s ∈ (0, 1), of a symmetric and uniformly elliptic second order operator and Caputo fractional time derivative of order γ ∈ (1, 2]. Since it has been shown useful for the design of numerical techniques for related problems, we also consider a quasi–stationary elliptic problem that comes from the realization of the spatial fractional diffusion as the Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on a semi–infinite cylinder. We provide existence and uniqueness results together with energy estimates for both problems. In addition, we derive regularity estimates both in time and space; the time–regularity results show that the usual assumptions made in the numerical analysis literature are problematic. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the large time behavior of compact support of the potential for a Camassa-Holm-type equation with nonlinearities of degree k+1 if the compactly supported initial potential keeps its sign. Moreover, persistence property in weighted Sobolev spaces is also investigated. [ABSTRACT FROM AUTHOR]

We propose and analyze a reliable and efficient a posteriori error estimator for a control-constrained linear-quadratic optimal control problem involving Dirac measures; the control variable corresponds to the amplitude of forces modeled as point sources. The proposed a posteriori error estimator is defined as the sum of two contributions, which are associated with the state and adjoint equations. The estimator associated with the state equation is based on Muckenhoupt weighted Sobolev spaces, while the one associated with the adjoint is in the maximum norm and allows for unbounded right hand sides. The analysis is valid for two and three-dimensional domains. On the basis of the devised a posteriori error estimator, we design a simple adaptive strategy that yields optimal rates of convergence for the numerical examples that we perform. [ABSTRACT FROM AUTHOR]

In this paper we are interested in the existence of a solution for the nonlinear degenerate elliptic equations Lu(x)+H(x,u,∇u)ω2 = f in the setting of the weighted Sobolev space W1,p0 (Ω,ω1,ω2 ), where H is a nonlinear term with natural growth with respect to ∇u and f ∊ L1(Ω) . [ABSTRACT FROM AUTHOR]

In this work we consider the initial value problem (IVP) associated to the Ostrovsky equations u t + ∂ x 3 u ± ∂ x − 1 u + u ∂ x u = 0 , x ∈ R , t ∈ R , u ( x , 0 ) = u 0 ( x ) . } We study the well-posedness of the IVP in the weighted Sobolev spaces Z ˙ s , s 2 : = { f ∈ H s ( R ) : ∂ x − 1 f ∈ L 2 ( R ) } ∩ L 2 ( | x | s d x ) , with 3 4 < s ≤ 1 . [ABSTRACT FROM AUTHOR]

Letbe a given domain and. We show that the validity of Poincaré inequalityimplies the solvability of degenerate elliptic eigenvalue problems, for sufficiently small. Our method exploits Ekeland’s Variational Principle and Palais–Smale sequences which are adapted to our quite general settings. We are assuming that the involved weightsbelong to the so-called-class, introduced by Kufner and Opic in 1984. Nonexistence of positive solutions of (S) for some choice of weights is also discussed. [ABSTRACT FROM PUBLISHER]

We use an approach based on pseudodifferential operators on Lie groupoids to study the double layer potentials on plane polygons. Let Ω be a simply connected polygon in R2. Denote by K the double layer potential operator on Ω associated with the Laplace operator Δ. We show that the operator K belongs to the groupoid C*-algebra that the first named author has constructed in an earlier paper (Carvalho and Qiao in Cent Eur J Math 11(1):27--54, 2013). By combining this result with general results in groupoid C*-algebras, we prove that the operators ±I +K are Fredholm between appropriate weighted Sobolev spaces, where I is the identity operator. Furthermore, we establish that the operators ±I +K are invertible between suitable weighted Sobolev spaces through techniques from Mellin transform. The invertibility of these operators implies a solvability result in weighted Sobolev spaces for the interior and exterior Dirichlet problems on Ω. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of solutions for strongly nonlinear degenerated elliptic unilateral problems of the form A(u)+g(x,u,∇u)+H(x,∇u)=f, where A is a Leray-Lions operator acting from W01,p(Ω,w) to its dual. On the nonlinear term g(x,s,ξ), we assume growth condition on ξ and a sign condition on s, while the convection term H(x,ξ) is only growing at most as |ξ|p-1. The right-hand side f belongs to L1(Ω). [ABSTRACT FROM AUTHOR]

The purpose of this work is to illustrate how the theory of Muckenhoupt weights, Muckenhoupt-weighted Sobolev spaces and the corresponding weighted norm inequalities can be used in the analysis and discretization of partial-differential-equation-constrained optimization problems. We consider a linear quadratic constrained optimization problem where the state solves a nonuniformly elliptic equation, a problem where the cost involves pointwise observations of the state and one where the state has singular sources, e.g., point masses. For all the three examples, we propose and analyse numerical schemes and provide error estimates in two and three dimensions. While some of these problems might have been considered before in the literature, our approach allows for a simpler, Hilbert-space-based analysis and discretization and further generalizations. [ABSTRACT FROM AUTHOR]

Tempered fractional diffusion equations (TFDEs) involving tempered fractional derivatives on the whole space were first introduced in Sabzikar et al. (J Comput Phys 293:14–28, 2015), but only the finite-difference approximation to a truncated problem on a finite interval was proposed therein. In this paper, we rigorously show the well-posedness of the models in Sabzikar et al. (2015), and tackle them directly in infinite domains by using generalized Laguerre functions (GLFs) as basis functions. We define a family of GLFs and derive some useful formulas of tempered fractional integrals/derivatives. Moreover, we establish the related GLF-approximation results. In addition, we provide ample numerical evidences to demonstrate the efficiency and “tempered” effect of the underlying solutions of TFDEs. [ABSTRACT FROM AUTHOR]

We are concerned with the numerical treatment of boundary integral equations by the adaptive wavelet boundary element method. In particular, we consider the second kind Fredholm integral equation for the double layer potential operator on patchwise smooth manifolds contained in ℝ3. The corresponding operator equations are treated by adaptive implementations that are in complete accordance with the underlying theory. The numerical experiments demonstrate that adaptive methods really pay off in this setting. The observed convergence rates fit together very well with the theoretical predictions based on the Besov regularity of the exact solution. [ABSTRACT FROM AUTHOR]

We prove the existence and uniqueness of solution to the nonhomogeneous degenerate elliptic PDE of second order with boundary data in weighted Orlicz-Slobodetskii space. Our goal is to consider the possibly general assumptions on the involved constraints: the class of weights, the boundary data, and the admitted coefficients. We also provide some estimates on the spectrum of our degenerate elliptic operator. [ABSTRACT FROM AUTHOR]

In this paper a class of infinite horizon optimal control problems with an isoperimetrical constraint, also interpreted as a budget constraint, is considered. Herein a linear both in the state and in the control dynamic is allowed. The problem setting includes a weighted Sobolev space as the state space. For this class of problems, we establish the necessary optimality conditions in form of a Pontryagin Type Maximum Principle including a transversality condition. The proved theoretical result is applied to a linear–quadratic regulator problem. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze a fractional spectral method for the time-fractional diffusion equation (TFDE). The main novelty of the method is approximating the solution by using a new class of generalized fractional Jacobi polynomials (GFJPs), also known as Müntz polynomials.We construct two efficient schemes using GFJPs for TFDE: one is based on the Galerkin formulation and the other on the Petrov--Galerkin formulation. Our theoretical or numerical investigation shows that both schemes are exponentially convergent for general right-hand side functions, even though the exact solution has very limited regularity (less than H¹). More precisely, an error estimate for the Galerkin-based approach is derived to demonstrate its spectral accuracy, which is then confirmed by numerical experiments. The spectral accuracy of the Petrov-- Galerkin-based approach is only verified by numerical tests without theoretical justification. Implementation details are provided for both schemes, together with a series of numerical examples to show the efficiency of the proposed methods. [ABSTRACT FROM AUTHOR]

We consider an optimal control problem that entails the minimization of a nondifferentiable cost functional, fractional diffusion as state equation and constraints on the control variable. We provide existence, uniqueness and regularity results together with first-order optimality conditions. In order to propose a solution technique, we realize fractional diffusion as the Dirichlet-to-Neumann map for a nonuniformly elliptic operator and consider an equivalent optimal control problem with a nonuniformly elliptic equation as state equation. The rapid decay of the solution to this problem suggests a truncation that is suitable for numerical approximation. We propose a fully discrete scheme: piecewise constant functions for the control variable and first-degree tensor product finite elements for the state variable.We derive a priori error estimates for the control and state variables. [ABSTRACT FROM AUTHOR]

We study properties of W 0 1 , p (R + , t β) — the completion of C 0 ∞ (R +) in the power-weighted Sobolev spaces W 1 , p (R + , t β) , where β ∈ R. Among other results, we obtain the analytic characterization of W 0 1 , p (R + , t β) for all β ∈ R. Our analysis is based on the precise study of the two trace operators: T r 0 (u) ≔ lim t → 0 u (t) and T r ∞ (u) ≔ lim t → ∞ u (t) , which leads to the analysis of the asymptotic behavior of functions from W 0 1 , p (R + , t β) near zero or infinity. The obtained statements can contribute to the proper formulation of Boundary Value Problems in ODEs, or PDEs with the radial symmetries. We can also apply our results to some questions in the complex interpolation theory, raised by Cwikel and Einav (2019), which we discuss within the particular case of Sobolev spaces W 1 , p (R + , t β). [ABSTRACT FROM AUTHOR]

In this paper we investigate the variable coefficient two-sided fractional diffusion, advection, reaction equations on a bounded interval. It is known that the fractional diffusion operator may lose coercivity due to the variable coefficient, which makes both the mathematical and numerical analysis challenging. To resolve this issue, we design appropriate test and trial functions to prove the inf-sup condition of the variable coefficient fractional diffusion, advection, reaction operators in suitable function spaces. Based on this property, we prove the well-posedness and regularity of the solutions, as well as analyze the Petrov–Galerkin approximation scheme for the proposed model. Numerical experiments are presented to substantiate the theoretical findings and to compare the behaviors of different models. [ABSTRACT FROM AUTHOR]

In this paper we are interested in the existence of solutions for the Dirichlet problem associated with degenerate nonlinear elliptic equations \[\begin{split}&-\sum_{j=1}^n D_j{\bigl[}{\omega}(x) {\cal A}_j(x, u, {\nabla}u){\bigr]} + b(x, u, {\nabla}u)\,{\omega}(x) + g(x)\,u(x)=\\&= f_0(x) - \sum_{j=1}^nD_jf_j(x) \quad{\rm on}\quad {\Omega}\end{split}\] in the setting of the weighted Sobolev spaces \({\rm W}_0^{1,p}(\Omega, \omega)\).