Zhuolin Li
We establish partial regularity for the $ω$-minimizers of quasiconvex functionals of power growth. A first-order partial regularity result of $BV$ $ω$-minimizers is obtained in the linear growth case under a Dini-type condition on $ω$. Only assuming the smallness of $ω$ near the origin, we show partial Hölder continuity in the subquadratic case by considering a normalised excess.
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We prove a partial regularity result for local minimizers of quasiconvex variational integrals with general growth. The main tool is an improved A-harmonic approximation, which should be interesting also for classical growth.
We prove the partial Hölder continuity for minimizers of quasiconvex functionals \[ \mathcal{F}({\bf u}) \colon =\int_Ω f(x,{\bf u},D{\bf u})\,\mathrm{d}x, \] where $f$ satisfies a uniform VMO condition with respect to the $x$-variable and is continuous with respect to ${\bf u}$. The growth condition with respect to the gradient variable is assumed a general one.
We prove the partial Hölder continuity on boundary points for minimizers of quasiconvex non-degenerate functionals \begin{equation*} \mathcal{F}({\bf u}) \colon =\int_Ω f(x,{\bf u},D{\bf u})\,\mathrm{d}x, \end{equation*} where $f$ satisfies a uniform VMO condition with respect to the $x$-variable, is continuous with respect to ${\bf u}$ and has a general growth with respect to the gradient variable.
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 establish an $\varepsilon$-regularity result for the derivative of a map of bounded variation that minimizes a strongly quasiconvex variational integral of linear growth, and, as a consequence, the partial regularity of such BV minimizers. This result extends the regularity theory for minimizers of quasiconvex integrals on Sobolev spaces to the context of maps of bounded variation. Previous partial regularity results for BV minimizers in the linear growth set-up were confined to the convex situation.
We show how to infer sharp partial regularity results for relaxed minimizers of degenerate, nonuniformly elliptic quasiconvex functionals, using tools from Nonlinear Potential Theory. In particular, in the setting of functionals with $(p,q)$-growth - according to the terminology of Marcellini [52] - we derive optimal local regularity criteria under minimal assumptions on the data.
We establish the first partial regularity result for local minima of strongly $\mathscr{A}$-quasiconvex integrals in the case where the differential operator $\mathscr{A}$ possesses an elliptic potential $\mathbb{A}$. As the main ingredient, the proof works by reduction to the partial regularity for full gradient functionals. Specialising to particular differential operators, the results in this paper thereby equally yield novel partial regularity theorems in the cases of the trace-free symmetric gradient, the exterior derivative or the div-curl-operator.
We establish $\mathrm{C}^{\infty}$-partial regularity results for relaxed minimizers of strongly quasiconvex functionals \begin{align*} \mathscr{F}[u;Ω]:=\int_ΩF(\nabla u)\,\mathrm{d} x,\qquad u\colonΩ\to\mathbb{R}^{N}, \end{align*} subject to a $q$-growth condition $|F(z)|\leq c(1+|z|^{q})$, $z\in\mathbb{R}^{N\times n}$, and natural $p$-mean coercivity conditions on $F\in\mathrm{C}^{\infty}(\mathbb{R}^{N\times n})$ for the basically optimal exponent range $1\leq p\leq q<\min\{\frac{np}{n-1},p+1\}$. With the $p$-mean coercivity condition being stated in terms of a strong quasiconvexity conditi…
We establish partial Hölder regularity for (local) generalised minimisers of variational problems involving strongly quasi-convex integrands of linear growth, where the full gradient is replaced by a first order homogeneous differential operator $\mathbb{A}$ with constant coefficients. Working under the assumption of $\mathbb{A}$ being $\mathbb{C}$-elliptic, this is achieved by adapting a method recently introduced by Gmeineder (Partial Regularity for Symmetric Quasiconvex Functionals on BD, J. Math. Pures Appl. 145 (2021), Issue 9, pp. 83--129) and Gmeineder & Kristensen (Partial regularity f…
A limiting case in partial regularity for quasiconvex functionals
该研究探讨非齐次拟凸变分积分局部极小元的梯度正则性，证明当右端项f属于临界Marcinkiewicz空间L(n,∞)时，梯度具有几乎处处BMO正则性，明确了该类泛函部分正则性的极限情形。
We establish the first partial regularity results for (strongly) symmetric quasiconvex functionals of linear growth on BD, the space of functions of bounded deformation. By Rindler's foundational work (Lower semicontinuity for integral functionals in the space of functions of bounded deformation via rigidity and Young measures, Arch. Ration. Mech. Anal. 202 (2011), no. 1, 63-113), symmetric quasiconvexity is the foremost notion as to sequential weak*-lower semicontinuity of functionals on BD. The overarching main difficulty here is Ornstein's Non-Inequality, implying that the BD-case is genuin…
We announce new existence and $\varepsilon$-regularity results for minimisers of the relaxation of strongly quasiconvex integrals that on smooth maps $u\colonΩ\subset\mathbb{R}^{n}\to\mathbb{R}^{N}$ are defined by $$u\mapsto \int_ΩF(\nabla^{k}u)dx.$$ The results cover the case of integrands $F$ with $(1,q)$-growth in the full range of exponents $1<q<\frac{n}{n-1}$ for which a measure representation of the relaxed functional is possible and the minimizers belong to the space $BV^k$ of maps whose $k$-th order derivatives are measures.
Partial regularity for minima of higher-order quasiconvex integrands with natural Orlicz growth
本文证明了一类满足自然Orlicz增长的高阶拟凸泛函极小值的部分正则性定理，无需被积函数二阶导数的定量估计，并推广至强局部极小值情形。
Partial regularity for $\mathbb{A}$-quasiconvex functionals with Orlicz growth
针对基于椭圆算子的微分表达式的一类泛函，在Orlicz增长和自然强拟凸性条件下，通过约化为全梯度部分正则性情形，证明了极小元的部分正则性。
Double phase quasiconvex functionals and their partial regularity theory
本文研究了退化非自治能量函数的最小化问题，通过拟凸性和正则性假设，建立梯度的Hölder正则性，利用A-调和近似和变分ϕ-调和近似方法。
In this paper the existence of an absolute minimizer for a functional \[ F(u,Ω) = \underset{x \in Ω}{\text{ess sup}} \, f (x, u(x), Du(x)) \] is proved by using Perron's method. The function is assumed to be quasiconvex and uniformly coercive. This completes the result by Champion, De Pascale and Prinari.
Asymptotically quasiconvex functionals with general growth conditions
本文在渐近 $φ$-拟凸性和 $ψ$-增长条件下，证明了变分法自治向量积分极小子的局部 $C^{1,α}$ 及稠密集上的 Lipschitz 正则性。
Regularity for Minimizers of Mixed-Smoothness Variational Integrals
该研究针对各向异性Sobolev空间上的拟凸变分积分极小元，建立了首个部分正则性结果，相关积分用于 pantographic 超材料的高阶梯度连续介质模型。
We consider the $L^\infty$-optimal mass transportation problem \[ \min_{Π(μ, ν)} γ-\mathrm{ess\,sup\,} c(x,y), \] for a new class of costs $c(x,y)$ for which we introduce a tentative notion of twist condition. In particular we study the conditions under which the infinitely-motonone minimizers are induced by a transportation map. We also state a uniqueness result for infinitely cyclically monotone Monge minimizers that corresponds to this class of cost functions. We compare the results to previous works.
Given a $C^2$ family of vector fields $X_1,...,X_m$ which induces a continuous Carnot-Carathéodory distance, we show that any absolute minimizer of a supremal functional defined by a $C^2$ quasiconvex Hamiltonian $f(x, z, p)$, allowing $z$-variable dependence, is a viscosity solution to the Aronsson equation $-\langle X(f(x, u(x), Xu(x))), D_p f(x, u(x), Xu(x))\rangle = 0.$
We consider functionals of the form $$\mathcal{F}(u):=\int_Ω\!F(x,u,\nabla u)\,\mathrm{d} x,$$ where $Ω\subseteq\mathbb{R}^n$ is open and bounded. The integrand $F\colonΩ\times\mathbb{R}^N\times\mathbb{R}^{N\times n}\to\mathbb{R}$ is assumed to satisfy the classical assumptions of a power $p$-growth and the corresponding strong quasiconvexity. In addition, $F$ is Hölder continuous with exponent $2β\in(0,1)$ in its first two variables uniformly with respect to the third variable, and bounded below by a quasiconvex function depending only on $z\in\mathbb{R}^{N\times n}$. We establish that strong…
Partial regularity for $\mathscr{A}$-quasiconvex variational problems of linear growth
本文研究了线性增长变分积分的极小解部分连续性，证明了在合适意义下，$\mathscr{A}$-拟凸积分算子的极小解具有部分正则性，涵盖了线性PDE算子和$\mathscr{A}$-自由变分情况。
We prove partial regularity for minimizers of quasiconvex functionals of the type $\int_Ωf(x,Du) dx$ with $p(x)$ growth with respect to the second variable. The proof is direct and uses a method of $A$-harmonic approximation.
In this paper we study the local regularity of almost minimizers of the functional \begin{equation*} J(u)=\int_Ω|\nabla u(x)|^2 +q^2_+(x)χ_{\{u>0\}}(x) +q^2_-(x)χ_{\{u<0\}}(x) \end{equation*} where $q_\pm \in L^\infty(Ω)$. Almost minimizers do not satisfy a PDE or a monotonicity formula like minimizers do (see \cite{AC}, \cite{ACF}, \cite{CJK}, \cite{W}). Nevertheless we succeed in proving that they are locally Lipschitz, which is the optimal regularity for minimizers.
We investigate properties of minimizers of a variational model describing the shape of charged liquid droplets. The model, proposed by Muratov and Novaga, takes into account the regularizing effect due to the screening of free counterionions in the droplet. In particular we prove partial regularity of minimizers, a first step toward the understanding of further properties of minimizers.
We study the partial regularity of minimizers for certain singular functionals in the calculus of variations, motivated by Ball and Majumdar's recent modification [BM] of the Landau-de Gennes energy functional.
We prove some variational analysis of regularity and weak convergence of nonlocal variational principle.
Regularity of two-phase free boundary minimizers in periodic media
本文研究周期介质中两相能量泛函极小元的正则性，建立了大尺度Lipschitz估计，并通过非退化极小元的平坦性改进，最终得到Liouville性质。
The repulsion strength at the origin for repulsive/attractive potentials determines the regularity of local minimizers of the interaction energy. In this paper, we show that if this repulsion is like Newtonian or more singular than Newtonian (but still locally integrable), then the local minimizers must be locally bounded densities (and even continuous for more singular than Newtonian repulsion). We prove this (and some other regularity results) by first showing that the potential function associated to a local minimizer solves an obstacle problem and then by using classical regularity results…
We revisit the question of regularity for minimizers of scalar autonomous integral functionals with so-called $(p,q)$-growth. In particular, we establish Lipschitz regularity under the condition $\frac{q}p<1+\frac{2}{n-1}$ for $n\geq3$ which improves a classical result due to Marcellini~[JDE'91].
Partial regularity and higher integrability for A-quasiconvex variational problems
本文研究了变分问题中最小解的部分连续性及更高积分性，考虑了p增长问题和线性常秩PDE算子，以及Dirichlet边界条件下的A-自由场。
In this paper, we study almost minimizers for the parabolic thin obstacle (or Signorini) problem with zero obstacle. We establish their $H^{σ,σ/2}$-regularity for every $0<σ<1$, as well as $H^{β,β/2}$-regularity of their spatial gradients on the either side of the thin space for some $0<β<1$. A similar result is also obtained for almost minimizers for the Signorini problem with variable Hölder coefficients.
We prove regularity properties in the vector valued case for minimizers of variational integrals of the form $$A(u) = \int_ΩA(x,u,Du) dx$$ where the integrand $A(x,u,Du)$ is not necessarily continuous respect to the variable $x,$ grows polinomially like $|ξ|^p,$ $p \geq 2.$
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。
We prove higher differentiability of bounded local minimizers to some widely degenerate functionals, verifying superquadratic anisotropic growth conditions. In the two dimensional case, we prove that local minimizers to a model functional are locally Lipschitz continuous functions, without any restriction on the anisotropy.
Motivated by the construction of time-periodic solutions for the three-dimensional Landau-Lifshitz-Gilbert equation in the case of soft and small ferromagnetic particles, we investigate the regularity properties of minimizers of the micromagnetic energy functional at the boundary. In particular, we show that minimizers are regular provided the volume of the particle is sufficiently small. The approach uses a reflection construction at the boundary and an adaption of the well-known regularity theory for minimizing harmonic maps into spheres.
We study optimal regularity and free boundary for minimizers of an energy functional arising in cohesive zone models for fracture mechanics. Under smoothness assumptions on the boundary conditions and on the fracture energy density, we show that minimizers are $C^{1, 1/2}$, and that near non-degenerate points the fracture set is $C^{1, α}$, for some $α\in (0, 1)$.
In this paper we consider minimizers for nonlocal energy functionals generalizing elastic energies that are connected with the theory of peridynamics \cite{Silling2000} or nonlocal diffusion models \cite{Rossi}. We derive nonlocal versions of the Euler-Lagrange equations under two sets of growth assumptions for the integrand. Existence of minimizers is shown for integrands with joint convexity (in the function and nonlocal gradient components). By using the convolution structure we show regularity of solutions for certain Euler-Lagrange equations. No growth assumptions are needed for the exist…
Regularity results for Hölder minimizers to functionals with non-standard growth
本文研究非自治泛函在$(p,q)$-增长和Besov系数假设下Hölder连续极小元的正则性，证明了其更高可积性和可微性。
Free boundary regularity for almost minimizers of the parabolic Signorini problem
本文针对抛物型Signorini问题中的几乎极小化子，通过建立Weiss型单调性公式和Almgren型频率公式，结合epiperimetric不等式，证明了正则自由边界点的优化增长及正则集的正则性。
We study the regularity of minimizers to the composite membrane problem in the plane (ie given a domain omega and a positive number A, smaller than the measure of omega, minimize the first Dirichlet eigenvalue for the Schrodinger operator with potential equal to a fixed multiple of the characteristic function of a subset D of omega, with measure A). We show that for minimizers, the boundary of D is analytic.
A Short Proof of Optimal Regularity for minimizers of the Alt-Phillips Problem
针对γ∈(0, 1)的 Alt - Phillips 自由边界问题极小值的最优正则性，给出自包含简短证明，核心方法是采用源自[DS20]的二分法论证。
We investigate minimizers defined on a bounded domain $Ω$ in $\mathbb{R}^2$ for singular constrained energy functionals that include Ball and Majumdar's modification of the Landau-de Gennes Q-tensor model for nematic liquid crystals. We prove regularity of minimizers with finite energy and show that their range on compact subdomains of $Ω$ does not intersect the boundary of the constraining set.
Regularity improvement for the minimizers of the two-dimensional Griffith energy
本文证明二维Griffith能量连通极小化子的奇异集维数小于1，并提高对称梯度的可积性。
We study the regularity of solutions to a nonlocal variational problem, which is related to the image denoising model, and we show that, in two dimensions, minimizers have the same Hölder regularity as the original image. More precisely, if the datum is (locally) $β$-Hölder continuous for some $β\in (1-s,\,1]$, where $s \in (0,1)$ is a parameter related to the nonlocal operator, we prove that the solution is also $β$-Hölder continuous.
We prove that, given~$p>\max\left\{\frac{2n}{n+2},1\right\}$, the nonnegative almost minimizers of the nonlinear free boundary functional $$ J_p(u,Ω):=\int_Ω\Big( |\nabla u(x)|^p+χ_{\{u>0\}}(x)\Big)\,dx$$ are Lipschitz continuous.
We prove local Lipschitz regularity for bounded minimizers of functionals with nonstandard $p,q$-growth with the source term in the Lorentz space $L(N,1)$ under the restriction $q<p+1+p\,\min\left\{\frac 1N,\frac{2(p-1)}{Np-2p+2}\right\}$. This extends the recent work by Beck-Mingione to bounded minimizers under weaker hypothesis and is sharp for some special ranges of $p$, $q$ and $N$.
We investigate minimizers defined on a bounded domain in $\mathbb{R}^2$ for the Maier--Saupe Q--tensor energy used to characterize nematic liquid crystal configurations. The energy density is singular, as in Ball and Mujamdar's modification of the Ginzburg--Landau Q--tensor model, so as to constrain the competing states to have eigenvalues in the closure of a physically realistic range. We prove that minimizers are regular and in several model problems we are able to use this regularity to prove that minimizers have eigenvalues strictly within the physical range.
The functionals of double phase type \[ \mathcal{H} (u):= \int \left(|Du|^{p} + a(x)|Du|^{q} \right) dx, ( q > p > 1, a(x)\geq 0) \] are introduced in the epoch-making paper by Colombo-Mingione for constants $p$ and $q$, and investigated by them and Baroni. They obtained sharp regularity results for minimizers of such functionals. In this paper we treat the case that the exponents are functions of $x$ and partly generalize their regularity results.
Large scale regularity of almost minimizers of the one-phase problem in periodic media
该研究针对周期介质中的单相自由边界能量泛函，证明其极小元与几乎极小元满足大尺度Lipschitz估计及自由边界平坦蕴含Lipschitz估计的性质，采用De Silva和Savin提出的均匀介质相关技术推导。
We study almost-minimizers of anisotropic surface energies defined by a Hölder continuous matrix of coefficients acting on the unit normal direction to the surface. In this generalization of the Plateau problem, we prove almost-minimizers are locally Hölder continuously differentiable at regular points and give dimension estimates for the size of the singular set. We work in the framework of sets of locally finite perimeter and our proof follows an excess-decay type argument.
In this paper we present a new proof of the sufficiency theorem for strong local minimizers concerning $C^1$-extremals at which the second variation is strictly positive. The results are presented in the quasiconvex setting, in accordance with the original statement by Grabovsky and Mengesha (2009). The strategy that we follow relies on a decomposition theorem that allows to split a sequence of variations into its oscillating and its concentrating parts, as well as on a sufficiency result according to which smooth extremals are spatially-local minimizers. Furthermore, we prove partial regulari…
We establish Maximum Principles which apply to vectorial approximate minimizers of the general integral functional of Calculus of Variations. Our main result is a version of the Convex Hull Property. The primary advance compared to results already existing in the literature is that we have dropped the quasiconvexity assumption of the integrand in the gradient term. The lack of weak Lower Semicontinuity is compensated by introducing a nonlinear convergence technique, based on the approximation of the projection onto a convex set by reflections and on the invariance of the integrand in the gradi…
The Singular Set of Minima of Multiple Integrals
研究拟凸积分局部极小值的奇异集，证明其豪斯多夫维数严格小于环境维数。
We prove regularity results for the unique minimizer of the total variation functional, currently used in image processing analysis since the work by L. Rudin, S. Osher and E. Fatemi. In particular we show that if the source term $f$ is locally, respectively globally, Lipschitz, then the solution has the same regularity with local, respectively global, Lipschitz norm estimated accordingly. The result is proved in any dimension and for any (regular) domain. So far we extend a similar result proved earlier by V. Caselles, A. Chambolle and M. Novaga for dimension $N\leq 7$ and (in case of the glo…
Existence and Regularity of Minimizers for a Plateau Approximation Problem
本文研究Plateau近似问题极小值的存在性与正则性，推广了一维Steiner问题的近似结果到高维表面。
In this paper, we study the relaxed energy for biharmonic maps from a $m$-dimensional domain into spheres. By an approximation method, we prove the existence of a minimizer of the relaxed energy of the Hessian energy, and that the minimizer is biharmonic and smooth outside a singular set $Σ$ of finite $(m-4)$-dimensional Hausdorff measure. Moreover, when $m=5$, we prove that the singular set $Σ$ is 1-rectifiable.
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.
The local Burkholder functional, quasiconvexity and Geometric Function Theory
本文证明局部Burkholder泛函是拟凸的，并在p趋于2时构造出正行列式矩阵集上的非多凸拟凸泛函，同时建立面积公式的精确推广，从而得到退化非多凸泛函极小化元的存在性。
Lipschitz regularity of almost-minimizers in one-phase problems with generalized Orlicz growth
研究了单相问题中几乎极小值解的Lipschitz正则性，针对具有广义Orlicz增长的Alt-Caffarelli型泛函建立了最优局部正则性结果。
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