Francesca Angrisani
We obtain local Lipschitz regularity for minima of autonomous integrals in the calculus of variations, assuming $q$-growth hypothesis and $W^{1,p}$-quasiconvexity only asymptotically, both in the sub-quadratic and the super-quadratic case.
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We consider regularity issues for minima of non-autonomous functionals in the Calculus of Variations exhibiting non-uniform ellipticity features. We provide a few sharp regularity results for local minimizers that also cover the case of functionals with nearly linear growth. The analysis is carried out provided certain necessary approximation-in-energy conditions are satisfied. These are related to the occurrence of the so-called Lavrentiev phenomenon that that non-autonomous functionals might exhibit, and which is a natural obstruction to regularity. In the case of vector valued problems we c…
We prove an a priori estimate for the second derivatives of local minimizers of integral functionals of calculus of variation 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. The novelty here is that we deal with integrands satisfying subquadratic growth conditions with respect to gradient variable.
Regularity for Minimizers of non Autonomous Singular Functionals with Anisotropic Growth
研究非自治凸积分泛函在次二次各向异性增长条件下局部最小值的更高可微性，通过指数间隙和能量密度振荡的Sobolev假设得到结果。
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.
Regularity for minimizers of degenerate, non-autonomous, orthotropic integral functionals
本文研究了退化、非自治、各向异性积分泛函最小值的正则性问题，通过证明局部有界最小值的更高可微性，拓展了非自治泛函的正则性理论。
Higher Differentiability of Minimizers for Non-Autonomous Orthotropic Functionals
本文研究了非自治各向异性积分泛函极小元的更高可微性，在系数满足适当Sobolev正则性且对指数施加依赖于维度和系数正则性的间隙界条件下，证明了其主要结果。
This paper is devoted to the autonomous Lagrange problem of the calculus of variations with a discontinuous Lagrangian. We prove that every minimizer is Lipschitz continuous if the Lagrangian is coercive and locally bounded. The main difference with respect to the previous works in the literature is that we do not assume that the Lagrangian is convex in the velocity. We also show that, under some additional assumptions, the DuBois-Reymond necessary condition still holds in the discontinuous case. Finally, we apply these results to deduce that the value function of the Bolza problem is locally …
On the singular set of $\operatorname{BV}$ minimizers for non-autonomous functionals
本文研究非自治凸变分泛函的BV极小化子的正则性，在椭圆性和Hölder连续条件下，证明了梯度更高可积性并给出了奇异集的Hausdorff维数上界。
Regularity results for minimizers of non-autonomous integral functionals
本研究在(p,q)增长条件下推导非自治积分泛函极小元的高阶分数阶可微性，放宽了对函数g的可微性要求，且仅需f在无穷远满足严格凸性与增长条件即可成立，丰富了变分法正则性理论。
In this paper we study the asymptotic behavior of solutions for a non-local non-autonomous scalar quasilinear parabolic problem in one space dimension. Our aim is to give a fairly complete description of the the forwards asymptotic behavior of solutions for models with Kirchoff type diffusion. In the autonomous we use the gradient structure of the model, some symmetry properties of solutions and develop comparison results to obtain a sequence of bifurcations of equilibria analogous to that seen in the model with local diffusivity. We give conditions so that the autonomous problem admits at mos…
We consider a reaction-diffusion equation with a nonlinear term of the Fisher-KPP type, depending on time $t$ and admitting two limits as $t\to\pm\infty$. We derive the set of admissible asymptotic past and future speeds of transition fronts for such equation. We further show that any transition front which is non-critical as $t\to-\infty$ always admits two asymptotic past and future speeds. We finally describe the asymptotic profiles of the non-critical fronts as $t\to\pm\infty$.
We establish local $C^{1,α}$-regularity for some $α\in(0,1)$ and $C^α$-regularity for any $α\in(0,1)$ of local minimizers of the functional \[ v\ \mapsto\ \int_Ωϕ(x,|Dv|)\,dx, \] where $ϕ$ satisfies a $(p,q)$-growth condition. Establishing such a regularity theory with sharp, general conditions has been an open problem since the 1980s. In contrast to previous results, we formulate the continuity requirement on $ϕ$ in terms of a single condition for the map $(x,t)\mapsto ϕ(x,t)$, rather than separately in the $x$- and $t$-directions. Thus we can obtain regularity results for functionals without…
We construct pullback attractors to the weak solutions of the three-dimensional Dirichlet problem for the incompressible Navier-Stokes equations in the case when the external force may become unbounded as time goes to plus or minus infinity.
In this paper, we study the Wasserstein gradient flow structure of the porous medium equation. We prove that, for the case of $q$-Gaussians on the real line, the functional derived by the JKO-discretization scheme is asymptotically equivalent to a rate-large-deviation-like functional. The result explains why the Wasserstein metric as well as the combination of it with the Tsallis-entropy play an important role.
We consider asymptotically autonomous semilinear parabolic equations u_t + Au = f(t,u). Suppose that $f(t,.)\to f^\pm$ as $t\to\pm\infty$, where the semiflows induced by \label{eq:140602-1511} u_t + Au = f^\pm(u) \tag{*} are gradient-like. Under certain assumptions, it is shown that generically with respect to a perturbation $g$ with $g(t)\to 0$ as $|t|\to\infty$, every solution of u_t + Au = f(t,u) + g(t) is a connection between equilibria $e^\pm$ of \eqref{eq:140602-1511} with $m(e^-)\geq m(e^+)$. Moreover, if the Morse indices satisfy $m(e^-) = m(e^+)$, then $u$ is isolated by linearization…
We study the asymptotic behavior of a family of functionals which penalize a short-range interaction of convolution type between a finite perimeter set and its complement. We first compute the pointwise limit and we obtain a lower estimate on more regulars sets. Finally, some examples are discussed.
We prove partial regularity for minimizers of vectorial integrals of the Calculus of Variations, with general growth condition, imposing quasiconvexity assumptions only in an asymptotic sense.
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.
On global absence of Lavrentiev gap for functionals with (p,q)-growth
本文研究具有(p,q)-增长的凸向量值泛函，在适当限制(p,q)及最小正则性假设下证明Lavrentiev现象直至边界均不发生，覆盖自治与非自治情形及多种增长界条件。
The Ginzburg-Landau functional is a phase transition model which is suitable for clustering or classification type problems. We study the asymptotics of a sequence of Ginzburg-Landau functionals with anisotropic interaction potentials on point clouds $Ψ_n$ where $n$ denotes the number data points. In particular we show the limiting problem, in the sense of $Γ$-convergence, is related to the total variation norm restricted to functions taking binary values; which can be understood as a surface energy. We generalize the result known for isotropic interaction potentials to the anisotropic case an…
In this paper the study of a nonlocal second order Cahn-Hilliard-type singularly perturbed family of functions is undertaken. The kernels considered include those leading to Gagliardo fractional seminorms for gradients. Using Gamma convergence the integral representation of the limit energy is characterized leading to an anisotropic surface energy on interfaces separating different phases.
Lipschitz regularity for a priori bounded minimizers of integral functionals with nonstandard growth
该研究在增长指数与椭圆性指数之差满足严格界的条件下，建立了满足非标准增长条件的非自治积分泛函先验有界局部极小元的Lipschitz正则性。
In this paper we firstly study the limit of minimizers of the fractional $W^{s,p}$-norms as $p\rightarrow+\infty$ in De Giorgi sense. In particular, we analyzed the $Γ$-convergence of non-homogeneous Dirichlet boundary problem for fractional $p$-Laplacian in this approximation process, and proved that as $p\rightarrow+\infty$ the minimizers of fractional $p$-Laplacian with Dirichlet boundary $Γ$-converges to a minimizer of Hölder $\infty$-Laplacian under the same Dirichlet boundary condition. On the other hand, we first investigate the asymptotic behaviour of non-homogeneous fractional $p$-fun…
In this paper, we prove the existences of pullback attractors in $L^{p}(\mathbb{R}^N)\times L^{2}(\mathbb{R}^N)$ for stochastic Fitzhugh-Nagumo system driven by both additive noises and deterministic non-autonomous forcings. The nonlinearity is polynomial like growth with exponent $p-1$. The asymptotic compactness for the cocycle in $L^{p}(\mathbb{R}^N)\times L^{2}(\mathbb{R}^N)$ is proved by using asymptotic a priori method, where the plus and minus signs of the nonlinearity at large value are not required.
Existence and asymptotic autonomous robustness of random attractors for three-dimensional stochastic globally modified Navier-Stokes equations on unbounded domains
本文研究了三维随机全局修正纳维-斯托克斯方程在无界领域中随机吸引子的存在性及渐近自抗扰性，通过证明解的回溯均匀性质，首次实现了该领域的相关理论突破。
Taking up a variational viewpoint, we present some nonlocal-to-local asymptotic results for various kinds of integral functionals. The content of the thesis comprises the contributions first appeared in some research papers in collaboration with J. Berendsen, A. Cesaroni, A. Chambolle, and M. Novaga.
In this paper, some global existence and uniform asymptotic stability results for fractional functional differential equations are proved. It is worthy mentioning that when $α=1$ the initial value problem (1.1) reduces to a classical dissipative differential equation with delays in [4]
We study the asymptotic behaviour of a properly normalized time changed Wiener processes. The time change reflects the fact that we consider the Laplace operator (which generates a Wiener process) multiplied by a possibly degenerate state-space dependent intensity $λ(x)$. Applying a functional limit theorem for the superposition of stochastic processes, we prove functional limit theorems for the normalized time changed Wiener process. The normalization depends on the asymptotic behaviour of the intensity function $λ$. One of the possible limits is a skew Brownian motion.
We consider adaptations of the Mumford-Shah functional to graphs. These are based on discretizations of nonlocal approximations to the Mumford-Shah functional. Motivated by applications in machine learning we study the random geometric graphs associated to random samples of a measure. We establish the conditions on the graph constructions under which the minimizers of graph Mumford-Shah functionals converge to a minimizer of a continuum Mumford-Shah functional. Furthermore we explicitly identify the limiting functional. Moreover we describe an efficient algorithm for computing the approximate …
The asymptotic behavior of an anisotropic Cahn-Hilliard functional with prescribed mass and Dirichlet boundary condition is studied when the parameter epsilon that determines the width of the transition layers tends to zero. The double-well potential is assumed to be even and equal to |s-1|^b near s=1, with 1<b<2. The first order term in the asymptotic development by Gamma-convergence is well-known, and is related to a suitable anisotropic perimeter of the interface. Here it is shown that, under these assumptions, the second order term is zero, which gives an estimate on the rate of convergenc…
We give a short description of the proof of asymptotic-completeness for NLS-type equations, including time dependent potential terms, with radial data in three dimensions. We also show how the method applies for the two-body Quantum Scattering case.
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