Variational Formulation Of Problem Research Articles

Stochastic processes associated to multidimensional parabolic transmission problems in divergence form

In this note we define and study the stochastic process $X$ in link with a parabolic transmission operator $(A,D(A))$ in divergence form. The transmission operator involves a diffraction condition along a transmission boundary. To that aim we gather and clarify some results coming from the theory of Dirichlet forms as exposed in [6] and [14] for general divergence form operators. We show that $X$ is a semimartingale and that it is solution of a stochastic differential equation involving partial reflections in the co-normal directions along the transmission boundary.

Helical vortices with small cross-section for 3D incompressible Euler equation

In this article, we construct traveling-rotating helical vortices with small cross-section to the 3D incompressible Euler equations in an infinite pipe, which tend asymptotically to singular helical vortex filament evolved by the binormal curvature flow. The construction is based on studying a general semilinear elliptic problem in divergence form{−ε2div(K(x)∇u)=(u−q|ln⁡ε|)+p,x∈Ω,u=0,x∈∂Ω, for small values of ε. Helical vortex solutions concentrating near several helical filaments with polygonal symmetry are also constructed.

On criticality theory for elliptic mixed boundary value problems in divergence form

The paper is devoted to the study of positive solutions of a second-order linear elliptic equation in divergence form in a domain [Formula: see text] that satisfy an oblique boundary condition on a portion of [Formula: see text]. First, we study weak solutions for the degenerate mixed boundary value problem [Formula: see text] where [Formula: see text] is a bounded Lipschitz domain, [Formula: see text] is a relatively open portion of [Formula: see text], and [Formula: see text] is an oblique (Robin) boundary operator defined on [Formula: see text] in a weak sense. In particular, we discuss the unique solvability of the above problem, the existence of a principal eigenvalue, and the existence of a minimal positive Green function. Then we establish a criticality theory for positive weak solutions of the operator [Formula: see text] in a general domain [Formula: see text] with no boundary condition on [Formula: see text] and no growth condition at infinity. The paper extends results obtained by Pinchover and Saadon for classical solutions of such a problem, where stronger regularity assumptions on the coefficients of [Formula: see text], and the boundary [Formula: see text] are assumed.

An SBV relaxation of the Cross-Newell energy for modeling stripe patterns and their defects

<p style='text-indent:20px;'>We investigate stripe patterns formation far from threshold using a combination of topological, analytic, and numerical methods. We first give a definition of the mathematical structure of 'multi-valued' phase functions that are needed for describing layered structures or stripe patterns containing defects. This definition yields insight into the appropriate 'gauge symmetries' of patterns, and leads to the formulation of variational problems, in the class of special functions with bounded variation, to model patterns with defects. We then discuss approaches to discretize and numerically solve these variational problems. These energy minimizing solutions support defects having the same character as seen in experiments.</p>

Convergence estimates of finite elements for a class of quasilinear elliptic problems

This paper is concerned with conforming finite element discretizations for quasilinear elliptic problems in divergence form, of a class that generalizes the p-Laplace equation and allows to show existence and uniqueness of the continuous and discrete problems. We derive discretization error estimates under general regularity assumptions for the solution and using high order polynomial spaces, resulting in convergence rates that are then verified numerically. A key idea of this error analysis is to consider carefully the relation between the natural W1,p-seminorm and a specific quasinorm introduced in the literature. In particular, we are able to derive interpolation estimates in this quasinorm from known interpolation estimates in the W1,p-seminorm. We also give a simplified proof of known near-best approximation results in W1,p-seminorm starting from the corresponding result in the mentioned quasinorm.

Finite Element Calculation of the Linear Elasticity Problem for Biomaterials with Fractal Structure

Aims: The aim of this study was to develop the mathematical models of the linear elasticity theory of biomaterials by taking into account their fractal structure. This study further aimed to construct a variational formulation of the problem, obtain the main relationships of the finite element method to calculate the rheological characteristics of a biomaterial with a fractal structure, and develop application software for calculating the components of the stress-strain state of biomaterials while considering their fractal structure. The obtained results were analyzed. Background: The development of adequate mathematical models of the linear elasticity theory for biomaterials with a fractal structure is an urgent scientific task. Finding its solution will make it possible to analyze the rheological behavior of biomaterials exposed to external loads by taking into account the existing effects of memory, spatial non-locality, self-organization, and deterministic chaos in the material. Objective: The objective of this study was the deformation process of biomaterials with a fractal structure under external load. Methods: The equations of the linear elasticity theory for the construction of the mathematical models of the deformation process of biomaterials under external load were used. Mathematical apparatus of integro-differentiation of fractional order to take into account the fractal structure of the biomaterial was used. A variational formulation of the linear elasticity problem while taking into account the fractal structure of the biomaterial was formulated. The finite element method with a piecewise linear basis for finding an approximate solution to the problem was used. Results: The main relations of the linear elasticity problem, which takes into account the fractal structure of the biomaterial, were obtained. A variational formulation of the problem was constructed. The main relations of the finite-element calculation of the linear elasticity problem of a biomaterial with a fractal structure using a piecewise-linear basis are found. The main components of the stress-strain state of the biomaterial exposed to external loads are found. Conclusion: Using the mathematical apparatus of integro-differentiation of fractional order in the construction of the mathematical models of the deformation process of biomaterials with a fractal structure makes it possible to take into account the existing effects of memory, spatial non-locality, self-organization, and deterministic chaos in the material. Also, this approach makes it possible to determine the residual stresses in the biomaterial, which play an important role in the appearance of stresses during repeated loads.

Inverse potential problems in divergence form for measures in the plane

We study inverse potential problems with source term the divergence of some unknown (ℝ3-valued) measure supported in a plane;e.g., inverse magnetization problems for thin plates. We investigate methods for recovering a magnetizationμby penalizing the measure-theoretic total variation norm ∥μ∥TV, and appealing to the decomposition of divergence-free measures in the plane as superpositions of unit tangent vector fields on rectifiable Jordan curves. In particular, we prove for magnetizations supported in a plane thatTV-regularization schemes always have a unique minimizer, even in the presence of noise. It is further shown thatTV-norm minimization (among magnetizations generating the same field) uniquely recovers planar magnetizations in the following two cases: (i) when the magnetization is carried by a collection of sufficiently separated line segments and a set that is purely 1-unrectifiable; (ii) when a superset of the support is tree-like. We note that such magnetizations can be recoveredvia TV-regularization schemes in the zero noise limit by taking the regularization parameter to zero. This suggests definitions of sparsity in the present infinite dimensional context, that generate results akin to compressed sensing.

A mixed variational formulation of dynamic viscoelastic problems with adhesion and friction

The aim of this work is to study a class of dynamic contact problems coupling adhesion and friction between two viscoelastic bodies with nonlinear viscosity operators. The boundary conditions concern relaxed unilateral contact, pointwise friction and adhesion including possible recoverable behavior. A mixed variational formulation of these problems is given as a five-field evolution implicit equation coupled with a differential inclusion describing the evolution of the intensity of adhesion. Based on several estimates and a classical fixed point theorem for multivalued functions, the existence of a strong variational solution is proved.

Besov regularity for solutions of elliptic equations with variable exponents

AbstractWe establish the higher fractional differentiability of the solutions of a problem in divergence form of the type The main features consist in assuming that the partial map has ‐growth, the datum F is Besov regular and both the partial map and the function are Orlicz–Besov regular.

Sparse recovery for inverse potential problems in divergence form

We discuss recent results from [10] on sparse recovery for inverse potential problem with source term in divergence form. The notion of sparsity which is set forth is measure- theoretic, namely pure 1-unrectifiability of the support. The theory applies when a superset of the support is known to be slender, meaning it has measure zero and all connected components of its complement has infinite measure in ℝ3. We also discuss open issues in the non-slender case.

A Third-Order Accurate Wave Propagation Algorithm for Hyperbolic Partial Differential Equations

We extend LeVeque’s wave propagation algorithm, a widely used finite volume method for hyperbolic partial differential equations, to a third-order accurate method. The resulting scheme shares main properties with the original method, i.e., it is based on a wave decomposition at grid cell interfaces, it can be used to approximate hyperbolic problems in divergence form as well as in quasilinear form and limiting is introduced in the form of a wave limiter.

Topology Optimization – a Variational Formulation of the Problem and Example Application

A variational formulation of the topology optimization problem is presented. A strain energy functional, being an equivalent of compliance, was minimized while constraints were imposed on the body mass. A global mass constraint and a local constraint on the amount of mass accumulated in a single material point of the body were adopted. A penalization procedure was defined and implemented in the optimization process to speed up the latter. The procedure in the successive optimization process steps translocates mass within the design domain, from the less strained areas to the more strained ones. The optimization process was described as a series of sequences of topologies determined using various control parameters, including different threshold functions. This means that the optimization process is characterized by a sequence of objective functional values approaching a minimal value. Various functions updating Young’s modulus were considered. Primarily the updating method referred to as SIMP was adopted. Three ways of using the discrete strain energy value to update Young's modulus in the considered material point were taken into account. These were: the amount of energy accumulated in the preceding step, the sum of the amounts of energy from all the preceding steps and the average amount of energy from the last two steps. In order to ensure the global limiting condition a mass constancy satisfaction procedure was incorporated into the algorithm. The algorithm procedures are described in detail. Finally, the algorithm was used to analyze selected problem relating to the pavement structure and the structure of tall buildings.

On some parabolic problems with measure sources

Abstract One of the recent advances in the investigation of nonlinear parabolic equations with a measure as forcing term is a paper by F. Petitta in which it has been introduced the notion of renormalized solutions to the initial parabolic problem in divergence form. Here we continue the study of the stability of renormalized solutions to nonlinear parabolic equations with measures but from a different point of view: we investigate the existence and uniqueness of the following nonlinear initial boundary value problems with absorption term and a possibly sign-changing measure data { b ( u ) t - div ( a ( t , x , u , ∇ u ) ) + h ( u ) = μ in Q : = ( 0 , T ) × Ω , u = 0 on ( 0 , T ) × ∂ Ω , b ( u ) = b ( u 0 ) in Ω , \left\{ {\matrix{ {b{{\left( u \right)}_t} - {\rm{div}}\left( {a\left( {t,x,u,\nabla u} \right)} \right) + h\left( u \right) = \mu } \hfill & {{\rm{in}}Q: = \left( {0,T} \right) \times {\rm{\Omega }},} \hfill \cr {u = 0} \hfill & {{\rm{on}}\left( {0,T} \right) \times \partial {\rm{\Omega }},} \hfill \cr {b\left( u \right) = b\left( {{u_0}} \right)} \hfill & {{\rm{in}}\,{\rm{\Omega }},} \hfill \cr } } \right. where Ω is an open bounded subset of ℝ N , N ≥ 2, T > 0 and Q is the cylinder (0, T) × Ω, Σ = (0, T) × ∂Ω being its lateral surface, the operator is modeled on the p−Laplacian with p > 2 - 1 N + 1 p > 2 - {1 \over {N + 1}} , μ is a Radon measure with bounded total variation on Q, b is a C 1 −increasing function which satisfies 0 < b 0 ≤ b′(s) ≤ b 1 (for positive constants b 0 and b 1). We assume that b(u 0) is an element of L 1(Ω) and h : ℝ ↦ ℝ is a continuous function such that h(s) s ≥ 0 for every |s| ≥ L and L ≥ 0 (odd functions for example). The existence of a renormalized solution is obtained by approximation as a consequence of a stability result. We provide a new proof of this stability result, based on the properties of the truncations of renormalized solutions. The approach, which does not need the strong convergence of the truncations of the solutions in the energy space, turns out to be easier and shorter than the original one.

Modeling of capillary coaxial gap filling with viscous liquid

A variational formulation of the boundary value problem on the motion of a viscous incompressible fluid with a free surface and changing dynamic boundary angles is proposed. The mathematical description of the process is based on the equations of motion, continuity and natural boundary conditions on the free surface. The traditional feature of the mathematical model on the three-phase contact lines (LTPC) is eliminated by the slip condition. The boundary angle on the LTPC is included in the variational formulation of the problem by replacing the curvature of the free boundary by the Laplace - Beltrami operator and using integration in parts. To describe the dynamic conditions on the LTPC, linking the speed of the LTPC and the dynamic edge angles on the solid walls of the cylinders, the empirical Jiang ratio is used. The numerical solution of the problem is based on the mixed finite element method with approximation of the main variables of the problem (velocity and pressure vector) satisfying the compatibility condition (LBB - condition). In addition, we use a singular finite element and a discontinuous approximation for the pressure to reduce the pressure oscillations in the vicinity of the LTPC. The numerical implementation of the kinematic condition of the free surface motion is performed according to the predictor-corrector scheme. The algorithm is tested on the problems with an analytical solution. Numerical studies of the flow kinematics and the free surface behavior when filling the coaxial gap in terms of the determining parameters of the Reynolds numbers (Re) in the range from 0 to 5, Stokes (W) in the range from 0 to 300 and capillary number (Ca) in the range from 0.0001 to 10 are carried out. The influence of the main parameters of the problem and dynamic conditions on LTPC on the evolution and maximum deflection of the free boundary is shown. For slow filling conditions, gravitational and capillary forces have the greatest influence on the kinematic characteristics of the filling. In this case, the dominant influence on the evolution of the free surface of the capillary forces begins with the values of the capillary numbers less than CA < 0,1. Increasing the flow rate (Re > 1) leads to a significant increase in the deflection of the free surface.

ABOUT SEVERAL NUMERICAL AND SEMIANALYTICAL METHODS OF LOCAL STRUCTURAL ANALYSIS

Numerical or semianalytical solution of problems of structural mechanics with immense number of unknowns is time-consuming process. High-accuracy solution at all points of the model is not required normally, it is necessary to find only the most accurate solution in some pre-known domains. The choice of these domains is a priori data with respect to the structure being modelled. Designers usually choose domains with the so-called edge effect (with the risk of significant stresses that could lead to destruction of structures) and regions which are subject to specific operational requirements. Stress-strain state in such domains is important. Wavelets provide effective and popular tool for local structural analysis. Operational and variational formulations of problems of structural mechanics with the use of method of extended domain are presented. After discretization and obtaining of governing equations, problems are transformed to a multilevel space by multilevel wavelet transform. Discrete wavelet basis is used and corresponding direct and inverse algorithms of transformations are performed. Due to special algorithms of averaging, reduction of the problems is provided. Wavelet-based methods allows reducing the size of the problems and obtaining accurate results in selected domains simultaneously. These are rather efficient methods for evaluation of local phenomenon in structures.

Gradient Weighted Norm Inequalities for Linear Elliptic Equations with Discontinuous Coefficients

Local and global weighted norm estimates involving Muckenhoupt weights are obtained for gradient of solutions to linear elliptic Dirichlet boundary value problems in divergence form over a Lipschitz domain $$\Omega $$ . The gradient estimates are obtained in weighted Lebesgue and Lorentz spaces, which also yield estimates in Lorentz–Morrey spaces as well as Holder continuity of solutions. The significance of the work lies on its applicability to very weak solutions (that belong to $$W^{1,p}_{0}(\Omega )$$ for some $$p>1$$ but not necessarily in $$W^{1,2}_{0}(\Omega )$$ ) to inhomogeneous equations with coefficients that may have discontinuities but have a small mean oscillation. The domain is assumed to have a Lipschitz boundary with small Lipschitz constant and as such allows corners. The approach implemented makes use of localized sharp maximal function estimates as well as known regularity estimates for very weak solutions to the associated homogeneous equations. The estimates are optimal in the sense that they coincide with classical weighted gradient estimates in the event the coefficients are continuous and the domain has smooth boundary.

Existence of renormalized solutions to quasilinear elliptic problems with general measure data

We consider a Dirichlet problem in divergence form with variable growth. We obtain existence of a renormalized solution for general measure data whose model is $$\begin{aligned} - \text {div } a(x,\nabla u)= & {} \mu \quad \text {in } \Omega \\ u= & {} 0 \quad \text {on }\partial \Omega , \end{aligned}$$ for any, possibly general, Radon measure $$\mu $$ with bounded total variation on $$\Omega $$ . The proofs rely crucially on a priori estimates.

On correctors for linear elliptic homogenization in the presence of local defects: The case of advection–diffusion

We follow-up on our works devoted to homogenization theory for linear second-order elliptic equations with coefficients that are perturbations of periodic coefficients. We have first considered equations in divergence form in [6–8]. We have next shown, in our recent work [9], using a slightly different strategy of proof than in our earlier works, that we may also address the equation −aij∂iju=f. The present work is devoted to advection–diffusion equations: −aij∂iju+bj∂ju=f. We prove, under suitable assumptions on the coefficients aij, bj, 1≤i,j≤d (typically that they are the sum of a periodic function and some perturbation in Lp, for suitable p<+∞), that the equation admits a (unique) invariant measure and that this measure may be used to transform the problem into a problem in divergence form, amenable to the techniques we have previously developed for the latter case.

Superconvergence of a class of expanded discontinuous Galerkin methods for fully nonlinear elliptic problems in divergence form

For fully nonlinear elliptic boundary value problems in divergence form, improved error estimates are derived in the frame work of a class of expanded discontinuous Galerkin methods. It is shown that the error estimate for the discrete flux in L 2 -norm is of order k + 1 , when piecewise polynomials of degree k ≥ 1 are used to approximate both potential as well as flux variables. Then, solving a discrete linear elliptic problem in each element locally, a suitable post-processing of the discrete potential is proposed and it is proved that the resulting post-processed potential converges with order of convergence k + 2 in L 2 -norm. By choosing stabilizing parameters appropriately, similar results are derived for the expanded HDG methods for nonlinear elliptic problems.

Global gradient estimates for non-uniformly elliptic equations

We consider a nonlinear and non-uniformly elliptic problem in divergence form on a bounded domain. The problem under consideration is characterized by the fact that its ellipticity rate and growth radically change with the position, which provides a model for describing a feature of strongly anisotropic materials. We establish the global Calderon–Zygmund type estimates for the distributional solution in the case that the boundary of the domain is of class \(C^{1,\beta }\) for some \(\beta >0\).