We consider a class of convex integral functionals composed of a term of linear growth in the gradient of the argument, and a fidelity term involving $L^2$ distance from a datum. Such functionals are known to attain their infima in the $BV$ space. Under the assumption that the domain of integration is convex, we prove that if the datum is in $W^{1,1}$, then the functional has a minimizer in $W^{1,1}$. In fact, the minimizer inherits $W^{1,p}$ regularity from the datum for any $p \in [1, +\infty]$. We also obtain a quantitative bound on the singular part of the gradient of the minimizer in the case that the datum is in $BV$. We infer analogous results for the gradient flow of the underlying functional of linear growth. We admit any convex integrand of linear growth.
A variational principle is introduced to provide a new formulation and resolution for several boundary value problems with a variational structure. This principle allows one to deal with problems well beyond the weakly compact structure. As a result, we study several super-critical semilinear Elliptic problems.
We consider a system of seminlinear parabolic variational inequalities with time-dependent convex obstacles. We prove the existence and uniqueness of its solution. We also provide a stochastic representation of the solution and show that it can be approximated by the penalization method. Our proofs are based upon probabilistic methods from the theory of Markov processes and the theory of backward stochastic differential equations.
By means of a recent variational technique, we prove the existence of radially monotone solutions to a class of nonlinear problems involving the $p$-Laplace operator. No subcriticality condition (in the sense of Sobolev spaces) is required.
We discuss variational problems on two-dimensional domains with energy densities of linear growth and with radially symmetric data. The smoothness of generalized minimizers is established under rather weak ellipticity assumptions. Further results concern the radial symmetry of solutions as well as a precise description of their behavior near the boundary.
The aim of the paper is to show that the solutions to variational problems with non-standard growth conditions satisfy a corresponding variational inequality without any smallness assumptions on the gap between growth and coercitivity exponents. Our results rely on techniques based on Convex Analysis that consist in establishing duality formulas and pointwise relations between minimizers and corresponding dual maximizers, for suitable approximating problems, that are preserved passing to the limit. In this respect we are able to show that the right class of competitors are the functions with f…
We prove existence of variational solutions for a class of nonlocal evolution equations whose prototype is the double phase equation \begin{align*} \partial_t u &+ \text{P.V.}\int_{\mathbb{R}^N} \frac{|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))}{|x-y|^{N+ps}}\\&+a(x,y)\frac{|u(x,t)-u(y,t)|^{q-2}(u(x,t)-u(y,t))}{|x-y|^{N+qr}} \,dy = 0. \end{align*} The approach of minimization of parameter-dependent convex functionals over space-time trajectories requires only appropriate convexity and coercivity assumptions on the nonlocal operator. As the parameter tends to zero, we recover variational solutions. Un…
We prove the optimal $W^{2,\infty}$ regularity for variational problems with convex gradient constraints. We do not assume any regularity of the constraints; so the constraints can be nonsmooth, and they need not be strictly convex. When the domain is smooth enough, we show that the optimal regularity holds up to the boundary. In this process, we also characterize the set of singular points of the viscosity solutions to some Hamilton-Jacobi equations. Furthermore, we obtain an explicit formula for the second derivative of these viscosity solutions; and we show that the second derivatives satis…
We prove local boundedness of generalized solutions to a large class of variational problems of linear growth including boundary value problems of minimal surface type and models from image analysis related to the procedure of TV-regularization occurring in connection with the denoising of images, which might even be coupled with an inpainting process. Our main argument relies on a Moser-type iteration procedure.
Variational inequalities, formulated on unknown dependent convex sets, are called quasi-variational inequalities (QVI). This paper is concerned with the abstract approach to a class of parabolic QVIs arising in many biochemical/mechanical problems, based on a compactness theorem for parabolic variational inequalities (cf. [9]). The prototype of our model for QVIs of parabolic type is formulated in a reflexive Banach space as the sum of the time-derivative operator under unknown convex constraints and a semimonotone operator, including a feedback system which selects a convex constraint. The ma…
In this paper we study the existence of solution for a class of variational inequality in whole $\mathbb{R}^N$ where the nonlinearity has a critical growth for $N \geq 2$. By combining a penalization scheme found in del Pino and Felmer [18] with a penalization method due to Bensoussan and Lions [9], we improve a recent result by Alves, Barros and Torres [1].
We establish local higher integrability and differentiability results for minimizers of variational integrals $$ \mathfrak{F}(v,Ω) = \int_Ω /! F(Dv(x)) \, dx $$ over $W^{1,p}$--Sobolev mappings $u \colon Ω\subset {\mathbb R}^n \to {\mathbb R}^N$ satisfying a Dirichlet boundary condition. The integrands $F$ are assumed to be autonomous, convex and of $(p,q)$ growth, but are otherwise not subjected to any further structure conditions, and we consider exponents in the range $1<p \leq q < p^{\ast}$, where $p^{\ast}$ denotes the Sobolev conjugate exponent of $p$.
In this note we present some results that were already conjectured in the work [9] by Bildhauer, Fuchs and Weickert, where they have investigated analytical aspects of coupled variational models with applications to mathematical imaging. Here we focus on variants of linear growth, which require a treatment in the framework of relaxation theory and convex analysis. Following basic ideas from [6] and [7], we establish existence and regularity of (dual-)solutions.
For proper minimizers of parabolic variational integrals with linear growth with respect to $|Du|$, we establish a necessary and sufficient condition for $u$ to be continuous at a point $(x_o,t_o)$, in terms of a sufficient fast decay of the total variation of $u$ about $(x_o,t_o)$ (see (1.4) below). These minimizers arise also as {proper} solutions to the parabolic $1$-laplacian equation. Hence, the continuity condition continues to hold for such solutions (§ 3).
We study the existence of solution for the following class of nonlocal problem, $$ -Δu +V(x)u =\Big( I_μ\ast F(x,u)\Big)f(x,u) \quad \mbox{in} \quad \mathbb{R}^2, $$ where $V$ is a positive periodic potential, $I_μ=\frac{1}{|x|^μ}$, $0<μ<2$ and $F(x,s)$ is the primitive function of $f(x,s)$ in the variable $s$. In this paper, by assuming that the nonlinearity $f(x,s)$ has an exponential critical growth at infinity, we prove the existence of solutions by using variational methods.
We give a survey of results regarding existence and regularity for autonomous functionals of linear growth that depend on the symmetric rather than the full gradients.
In this short note we prove the convexity of minimizers of some variational problem in the Gauss space. This proof is based on a geometric version of an older argument due to Korevaar.
Splitting-type variational problems \[ \int_Ω\sum_{i=1}^n f_i(\partial_i w) dx \to \min \] with superlinear growth conditions are studied by assuming \[ h_i(t) \leq f''_i(t) \leq H_i(t) \] with suitable functions $h_i$, $H_i$: $\mathbb{R} \to \mathbb{R}^+$, $i=1$, $\dots$ , $n$, measuring the growth and ellipticity of the energy density. Here, as the main feature, a symmetric behaviour like $h_i(t)\approx h_i(-t)$ and $H_i(t) \approx H_i(-t)$ for large $|t|$ is not supposed. Assuming quite weak hypotheses as above, we establish higher integrability of $|\nabla u|$ for local minimizers $u\in L^…
Variational problems of splitting-type with mixed linear-superlinear growth conditions are considered. In the twodimensional case the minimizing problem is given by \[ J [w] = \int_Ω \Big[f_1\big(\partial_1 w\big) + f_2\big(\partial_2 w\big)\Big] \,dx \to \min \] w.r.t. a suitable class of comparison functions. Here $f_1$ is supposed to be a convex energy density with linear growth, $f_2$ is supposed to be of superlinear growth, for instance to be given by a $N$-function or just bounded from below by a $N$-function. One motivation for this kind of problem located between the well known splitti…
We consider a Neumann problem for strictly convex variational functionals of linear growth. We establish the existence of minimisers among $\operatorname{W}^{1,1}$-functions provided that the domain under consideration is simply connected. Hence, in this situation, the relaxation of the functional to the space of functions of bounded variation, which has better compactness properties, is not necessary. Similar $\operatorname{W}^{1,1}$-regularity results for the corresponding Dirichlet problem are only known under rather restrictive convexity assumptions limiting its non-uniformity up to the bo…
In this paper we study the existence of ground state solution for an indefinite variational problem of the type $$ \left\{\begin{array}{l} -Δu+(V(x)-W(x))u=f(x,u) \quad \mbox{in} \quad \R^{N}, u\in H^{1}(\R^{N}), \end{array}\right. \eqno{(P)} $$ where $N \geq 2$, $V,W:\mathbb{R}^N \to \mathbb{R}$ and $f:\mathbb{R}^N \times \mathbb{R} \to \mathbb{R}$ are continuous functions verifying some technical conditions and $f$ possesses a critical growth. Here, we will consider the case where the problem is asymptotically periodic, that is, $V$ is $\mathbb{Z}^N$-periodic, $W$ goes to 0 at infinity and $…
We prove global Lipschitz regularity for a wide class of convex variational integrals among all functions in $W^{1,1}$ with prescribed (sufficiently regular) boundary values, which are not assumed to satisfy any geometrical constraint (as for example bounded slope condition). Furthermore, we do not assume any restrictive assumption on the geometry of the domain and the result is valid for all sufficiently smooth domains. The result is achieved with a suitable approximation of the functional together with a new construction of appropriate barrier functions.
A nested family of growing or shrinking planar domains is called a Laplacian growth process if the normal velocity of each domain's boundary is proportional to the gradient of the domain's Green function with a fixed singularity on the interior. In this paper we review the Laplacian growth model and its key underlying assumptions, so that we may consider a generalization to so-called elliptic growth, wherein the Green function is replaced with that of a more general elliptic operator--this models, for example, inhomogeneities in the underlying plane. In this paper we continue the development o…
We establish partial regularity of BD-minima for variational integrals of linear growth which depend on the symmetric gradients and satisfy a weak ellipticity condition. Since there is no Korn Inequality in the $L^{1}$-Setup, the result does not follow from the respective result on BV due to Anzellotti and Giaquinta.
We develop arguments on the critical point theory for locally Lipschitz functionals on Orlicz-Sobolev spaces, along with convexity and compactness techniques to investigate existence of solution of the multivalued equation $\displaystyle - Δ_Φ u \in \partial j(.,u) + λh \mbox{in} Ω$, where $Ω\subset {\bf R}^{N}$ is a bounded smooth domain, $Φ: {\r} \longrightarrow [0,\infty)$ is a suitable N-function, $Δ_Φ$ is the corresponding $Φ$-Laplacian, $λ> 0$ is a parameter, $h:Ω\rightarrow{\r}$ is integrable and $\partial j(., u)$ is the subdifferential of a function $j$ associated with critical growth…
We approximate the homogenization of fully nonlinear, convex, uniformly elliptic Partial Differential Equations in the periodic setting, using a variational formula for the optimal invariant measure, which may be derived via Legendre-Fenchel duality. The variational formula expresses $\bar H$ as an average of the operator against the optimal invariant measure, generalizing the linear case. Several nontrivial analytic formulas for $\bar H$ are obtained. These formulas are compared to numerical simulations, using both PDE and variational methods. We also perform a numerical study of convergence …
We revisit the problem of approximating minimizers of certain convex functionals subject to a convexity constraint by solutions of fourth order equations of Abreu type. This approximation problem was studied in previous works of Carlier-Radice (Approximation of variational problems with a convexity constraint by PDEs of Abreu type. Calc. Var. Partial Differential Equations. 58 (2019), no. 5, Art. 170) and the author (Singular Abreu equations and minimizers of convex functionals with a convexity constraint, arXiv:1811.02355v3, Comm. Pure Appl. Math., to appear), under the uniform convexity of b…
Regularity properties of solutions to variational problems are established for a broad class of strictly convex splitting-type energy densities of the principal form $f$: $\mathbb{R}^2 \to \mathbb{R}$, \[ f(ξ_1,ξ_2) = f_1\big( ξ_1 \big) + f_2\big( ξ_2 \big) \, , \] with linear growth. As a main result it is shown that, regardless of a corresponding property of $f_2$, the assumption ($t\in \mathbb{R}$) $c_1 (1+|t|)^{-μ_{1}} \le f_1''(t) \le c_2\, ,\quad 1 < μ_1 < 2\, ,$ is sufficient to obtain higher integrability of $\partial_1 u$ for any finite exponent. We also inculde a series of variants o…
Motivated by the direct method in the calculus of variations in $L^{\infty}$, our main result identifies the notion of convexity characterizing the weakly$^*$ lower semicontinuity of nonlocal supremal functionals: Cartesian level convexity. This new concept coincides with separate level convexity in the one-dimensional setting and is strictly weaker for higher dimensions. We discuss relaxation in the vectorial case, showing that the relaxed functional will not generally maintain the supremal form. Apart from illustrating this fact with examples of multi-well type, we present precise criteria f…
We prove weak duality between two recent convex relaxation methods for bounding the optimal value of a constrained variational problem in which the objective is an integral functional. The first approach, proposed by Valmorbida et al. (IEEE Trans. Automat. Control 61(6):1649--1654, 2016), replaces the variational problem with a convex program over sufficiently smooth functions, subject to pointwise non-negativity constraints. The second approach, discussed by Korda et al. (arXiv:1804.07565v1 [math.OC]), relaxes the variational problem into a convex program over scaled probability measures. We …
We show convexity of solutions to a class of convex variational problems in the Gauss and in the Wiener space. An important tool in the proof is a representation formula for integral functionals in this infinite dimensional setting, that extends analogous results valid in the classical Euclidean framework.
This article is devoted to characterize all possible effective behaviors of composite materials by means of periodic homogenization. This is known as a $G$-closure problem. Under convexity and $p$-growth conditions ($p>1$), it is proved that all such possible effective energy densities obtained by a $Γ$-convergence analysis, can be locally recovered by the pointwise limit of a sequence of periodic homogenized energy densities with prescribed volume fractions. A weaker locality result is also provided without any kind of convexity assumption and the zero level set of effective energy densities …
We prove the existence of solutions for a class of quasilinear problems involving variable exponents and with nonlinearity having critical growth. The main tool used is the variational method, more precisely, Ekeland's Variational Principle and the Mountain Pass Theorem.
In this paper, we prove multiplicity of solutions for a class of quasilinear problems in $ \mathbb{R}^{N} $ involving variable exponents and nonlinearities of concave-convex type. The main tools used are variational methods, more precisely, Ekeland's variational principle and Nehari manifolds.
This paper considers a general framework for the study of the existence of quasi-variational and variational solutions to a class of nonlinear evolution systems in convex sets of Banach spaces describing constraints on a linear combination of partial derivatives of the solutions. The quasi-linear operators are of monotone type, but are not required to be coercive for the existence of weak solutions, which is obtained by a double penalisation/regularisation for the approximation of the solutions. In the case of time-dependent convex sets that are independent of the solution, we show also the un…
We prove global $W^{1,q}(Ω,\mathbb{R}^m)$-regularity for minimisers of convex functionals of the form $\mathscr{F}(u)=\int_ΩF(x,Du)\mathrm{d} x$. $W^{1,q}(Ω,\mathbb{R}^m)$ regularity is also proven for minimisers of the associated relaxed functional. Our main assumptions on $F(x,z)$ are a uniform $α$-Hölder continuity assumption in $x$ and controlled $(p,q)$-growth conditions in $z$ with $q<\frac{(n+α)p}{n}$.
Monotone vector fields were introduced almost 40 years ago as nonlinear extensions of positive definite linear operators, but also as natural extensions of gradients of convex potentials. These vector fields are not always derived from potentials in the classical sense, and as such they are not always amenable to the standard methods of the calculus of variations. We describe here how the selfdual variational calculus developed recently by the author, provides a variational approach to PDEs and evolution equations driven by maximal monotone operators. To any such a vector field T on a reflexiv…
In this paper we are interested to prove the existence and concentration of ground state solution for the following class of problems $$ -Δu+V(x)u=A(εx)f(u), \quad x \in \R^{N}, \eqno{(P)_ε} $$ where $N \geq 2$, $ε>0$, $A:\R^{N}\rightarrow\R$ is a continuous function that satisfies $$ 0<\inf_{x\in\R^{N}}A(x)\leq\lim_{|x|\rightarrow+\infty}A(x)<\sup_{x\in\R^{N}}A(x)=A(0),\eqno{(A)} $$ $f:\R\rightarrow\R$ is a continuous function having critical growth, $V:\R^{N}\rightarrow\R$ is a continuous and $\Z^{N}$--periodic function with $0\notinσ(Δ+V)$. By using variational methods, we prove the existen…
We study the minimization of convex, variational integrals of linear growth among all functions in the Sobolev space $W^{1,1}$ with prescribed boundary values (or its equivalent formulation as a boundary value problem for a degenerately elliptic Euler--Lagrange equation). Due to insufficient compactness properties of these Dirichlet classes, the existence of solutions does not follow in a standard way by the direct method in the calculus of variations and in fact might fail, as it is well-known already for the non-parametric minimal surface problem. Assuming radial structure, we establish a ne…
We investigate the properties of certain elliptic systems leading, a~priori, to solutions that belong to the space of Radon measures. We show that if the problem is equipped with a so-called asymptotic Uhlenbeck structure, then the solution can in fact be understood as a standard weak solution, with one proviso: analogously as in the case of minimal surface equations, the attainment of the boundary value is penalized by a measure supported on (a subset of) the boundary, which, for the class of problems under consideration here, is the part of the boundary where a Neumann boundary condition is …
Utilizing a new variational principle that allows dealing with problems beyond the usual locally compactness structure, we study problems with a supercritical nonlinearity of the type $ -Δu + u= a(x) f(u)$ in $ Ω$ with $\partial_νu=0$ on $ \partial Ω$. Here $Ω$ is a bounded domain with certain symmetry assumptions. We find positive nontrivial solutions in the case of suitable supercritical nonlinearities $f$ by finding critical points of $I$ where \[ I(u)=\int_Ω\left\{ a(x) F^* \left( \frac{-Δu + u}{a(x)} \right) - a(x) F(u) \right\} dx, \] over the closed convex cone $K_m$ of nonnegative, sym…
Under $1<p\le 2$, this paper presents some old and new convexity/isoperimetry based inequalities for the variational $p$-capacity potentials on convex plane rings.
We are concerned with the study of the well-posedness of a nonlinear diffusion equation with a monotonically increasing multivalued time-dependent nonlinearity derived from a convex continuous potential having a superlinear growth to infinity. The results in this paper state that the solution of the nonlinear equation can be retrieved as the null minimizer of an appropriate minimization problem for a convex functional involving the potential and its conjugate. This approach, inspired by the Brezis-Ekeland variational principle, provides new existence results under minimal growth and coercivity…
We study homogenization by Gamma-convergence of periodic multiple integrals of the calculus of variations when the integrand can take infinite values outside of a convex set of matrices.
