We consider autonomous integral functionals of the form $\mathcal F[u]:=\int_Ωf(D u)\,dx$ with $u:Ω\to\mathbb R^N$ $N\geq1$, where the convex integrand $f$ satisfies controlled $(p,q)$-growth conditions. We establish higher gradient integrability and partial regularity for minimizers of $\mathcal F$ assuming $\frac{q}p<1+\frac2{n-1}$, $n\geq3$. This improves earlier results valid under the more restrictive assumption $\frac{q}p<1+\frac2{n}$.
We prove continuity up to the boundary of the minimizer of an obstacle problem and higher integrability of its gradient under generalized Orlicz growth. The result recovers similar results obtained in the special cases of polynomial growth, variable exponent growth and produces new results for Orlicz and double phase growth.
We prove higher summability for the gradient of minimizers of strongly convex integral functionals of the Calculus of Variations with (p,q)-Growth conditions in low dimension. Our procedure is set in the framework of Fractional Sobolev Spaces and renders the desired regularity as the result of an approximation technique relying on estimates obtained through a careful use of difference quotients.
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 $q<\frac{(n+α)p}{n}$.
We prove global $W^{1,q}(Ω,\mathbb{R}^N)$-regularity for minimisers of $\mathscr{F}(u)=\int_ΩF(x,\mathrm{D}u)\mathrm{d} x$ satisfying $u\geq ψ$ for a given Sobolev obstacle $ψ$. $W^{1,q}(Ω,\mathbb{R}^m)$ 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 natural $(p,q)$-growth conditions in $z$ with $q<\frac{(n+α)p}{n}$. In the autonomous case $F\equiv F(z)$ we can improve the gap to $q<\frac{np}{n-1}$, a result new even in the unconstrained case.
We prove higher integrability of the spatial gradient of weak solutions to parabolic systems with $ϕ$-growth, where $φ=φ(t)$ is a general Orlicz function. The parabolic systems need be neither degenerate nor singular. Our result is a generalized version of the one of J. Kinnunen and J. Lewis [Duke Math. J. 102 (2000), no. 2, 253--271] for the parabolic $p$-Laplace systems.
Weak Differentiability to Nonuniform Nonlinear Degenerate Elliptic Systems under $p,q$-growth Condition on the Heisenberg Group
School of Mathematics and Data Science, Shaanxi University of Science and Technology（陕西科技大学数学与数据科学学院）；School of Sciences, Xi’an University of Technology（西安理工大学理学院）
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 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.
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].
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.
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 study regularity results for nonlinear parabolic systems of $p$-Laplacian type with inhomogeneous boundary and initial data, with $p\in(\frac{2n}{n+2},\infty)$. We show bounds on the gradient of solutions in the Lebesgue-spaces with arbitrary large integrability exponents and natural dependences on the right hand side and the boundary data. In particular, we provide a new proof of the global non-linear Calderón-Zygmund theory for such systems. Our method makes use of direct estimates on the solution minus its boundary values and hence is considerably shorter than the available higher integr…
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.
We pursue the study of a model convex functional with orthotropic structure and nonstandard growth conditions, this time focusing on the sub-quadratic case. We prove that bounded local minimizers are locally Lipschitz. No restriction on the ratio between the highest and the lowest growth rates are needed. The result holds also in presence of a non-autonomous lower order term, under sharp integrability assumptions. Finally, we prove higher differentiability of bounded local minimizers, as well.
We prove a new integrability principle for gradient variational problems in $\mathbb{R}^2$, showing that solutions are explicitly parameterized by $κ$-harmonic functions, that is, functions which are harmonic for the laplacian with varying conductivity $κ$, where $κ$ is the square root of the Hessian determinant of the surface tension.
We mainly discuss superquadratic minimization problems for splitting-type variational integrals on a bounded Lipschitz domain $Ω\subset \mathbb{R}^2$ and prove higher integrability of the gradient up to the boundary by incorporating an appropriate weight-function measuring the distance of the solution to the boundary data. As a corollary, the local Hölder coefficient with respect to some improved Hölder continuity is quantified in terms of the function ${\rm dist}(\cdot,\partial Ω)$. The results are extended to anisotropic problems without splitting structure under natural growth and elliptici…
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…
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 …
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 …
We consider a class of integral functionals with convex integrand with respect to the gradient variable, assuming that the function that measures the oscillation of the integrand with respect to the x variable belongs to a suitable Sobolev space W^{1;q}. We prove a result of higer differentiability for the minimizers. We also infer a result of Lipschitz regularity of minimizers if q > n, and a result of higher integrability for the gradient if q = n. The novelty here is that we deal with integrands satisfying subquadratic growth conditions with respect to gradient variable.
