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We extend the De Giorgi iteration technique to the vectorial setting. For this we replace the usual scalar truncation operator by a vectorial shortening operator. As an application, we prove local boundedness for local and nonlocal nonlinear systems. Furthermore, we show convex hull properties, which are a generalization of the maximum principle to the case of systems.

We propose a systematic elementary approach based on trajectories to prove Poincar\'e inequalities for hypoelliptic equations with an arbitrary number of H\"ormander commutators, both in the local and in the non-local case. Our method generalises and simplifies the method introduced in Guerand-Mouhot (2022) in the local case with one commutator, and extended in Loher (2024) to the non-local case with one commutator. It draws inspiration from the paper Niebel-Zacher (2022), although we use different trajectories. We deduce Harnack inequalities and H\"older regularity along the line of the De Giorgi method. Our results recover those in Anceschi-Rebucci (2022) in the local case, and are new in the non-local case., Comment: 22 pages, 4 figures. Simplified the Lemma on the determinant of the Wronskian matrix in the construction of the trajectories (Lemma 7), and introduced the notion of solutions we work with

We give an alternative proof of the Schoen--Simon--Yau curvature estimates and associated Bernstein-type theorems (1975), and extend the original result by including the case of $6$-dimensional (stable minimal) immersions. The key step is an $\epsilon$-regularity theorem, that assumes smallness of the scale-invariant $L^2$ norm of the second fundamental form. Further, we obtain a graph description, in the Lipschitz multi-valued sense, for any stable minimal immersion of dimension $n\geq 2$, that may have a singular set $\Sigma$ of locally finite $\mathcal{H}^{n-2}$-measure, and that is weakly close to a hyperplane. (In fact, if $\mathcal{H}^{n-2}(\Sigma)=0$, the conclusion is strengthened to a union of smooth graphs.) This follows directly from an $\epsilon$-regularity theorem, that assumes smallness of the scale-invariant $L^2$ tilt-excess (verified when the hypersurface is weakly close to a hyperplane). A further direct consequence is that (for an immersed hypersurface of the type considered) any tangent cone supported on a hyperplane is the unique tangent. Specialising the multi-valued decomposition to the case of embeddings, we recover the Schoen--Simon theorem (1981). In both $\epsilon$-regularity theorems the relevant quantity (respectively, length of the second fundamental form and tilt function) solves a non-linear PDE on the immersed minimal hypersurface. The proof is carried out intrinsically (without linearising the PDE) by implementing an iteration method \`{a} la De Giorgi (from the linear De Giorgi--Nash--Moser theory). Stability implies estimates (intrinsic weak Caccioppoli inequalities) that make the iteration effective despite the non-linear framework. (In both $\epsilon$-regularity theorems the method gives explicit constants that quantify the required smallness.), Comment: 31 pp

We prove De Giorgi-Nash-Moser Theory using a geometric approach., Comment: 14 pages, 1 figure

The title is meant as way to honor two great mathematicians that, although never actually worked together, introduced concepts of convergence that perfectly match each other and very fruitfully interact: De Giorgi's $\Gamma$-convergence of lower semicontinuous functions and Gromov's convergence of geometric structures.

We present an extension of the classical De Giorgi class, and then we show that functions in this new class are locally bounded and locally H\"older continuous. Some applications are given. As a first application, we give a regularity result for local minimizers $u:\Omega \subset \mathbb R^4 \rightarrow \mathbb R^4$ of a special class of polyconvex functionals with splitting form in four dimensional Euclidean spaces. Under some structural conditions on the energy density, we prove that each component $u^\alpha$ of the local minimizer $u$ belongs to the generalized De Giorgi class, then one can derive that it is locally bounded and locally H\"older continuous. Our result can be applied to polyconvex integrals whose prototype is $$ \int_\Omega \Big(\sum_{\alpha =1}^4 |Du^\alpha|^p + \sum_{\beta =1}^6 |({\rm adj}_2 Du )^\beta | ^q +\sum_{\gamma =1}^4 |({\rm adj}_3 Du )^\gamma | ^r +|\det Du|^s \Big ) \mathrm {d}x $$ with suitable $p,q,r,s\ge 1$. As a second application, we consider a degenerate linear elliptic equation of the form $$ -\mbox {div} (a(x)\nabla u)=-\mbox {div}F, $$ with $0

In this paper, we consider the global well-posedness to the non-cutoff Boltzmann equation with soft potential in the $L^\infty$ setting. We show that when the initial data is close to equilibrium and the perturbation is small in $L^2 \cap L^\infty$ polynomial weighted space, the Boltzmann equation has a global solution in the weighted $L^2 \cap L^\infty$ space. The ingredients of the proof lie in strong averaging lemma, new polynomial weighted estimate for the non-cutoff Boltzmann equation and the $L^2$ level set Di Giorgi iteration method developed in \cite{AMSY2}. The convergence to the equilibrium state in both $L^2$ and $L^\infty$ spaces is also proved.

In 1991 De Giorgi conjectured that, given $\lambda >0$, if $\mu_\varepsilon$ stands for the density of the Allen-Cahn energy and $v_\varepsilon$ represents its first variation, then $\int [v_\varepsilon^2 + \lambda] d\mu_\varepsilon$ should $\Gamma$-converge to $c\lambda \mathrm{Per}(E) + k \mathcal{W}(\Sigma)$ for some real constant $k$, where $\mathrm{Per}(E)$ is the perimeter of the set $E$, $\Sigma=\partial E$, $\mathcal{W}(\Sigma)$ is the Willmore functional, and $c$ is an explicit positive constant. A modified version of this conjecture was proved in space dimensions $2$ and $3$ by R\"oger and Sch\"atzle, when the term $\int v_\varepsilon^2 \, d\mu_\varepsilon$ is replaced by $ \int v_\varepsilon^2 {\varepsilon}^{-1} dx$, with a suitable $k>0$. In the present paper we show that, surprisingly, the original De Giorgi conjecture holds with $k=0$. Further properties on the limit measures obtained under a uniform control of the approximating energies are also provided., Comment: 32 pages. Comments are welcome!

We derive quantitatively the Harnack inequalities for kinetic integro-differential equations. This implies H\"older continuity. Our method is based on trajectories and exploits a term arising due to the non-locality in the energy estimate. This permits to quantitatively prove the intermediate value lemma for the full range of non-locality parameter $s \in (0, 1)$. Our results recover the results from Imbert and Silvestre [22] for the inhomogeneous Boltzmann equation in the non-cutoff case. The paper is self-contained., Comment: 53 pages, 2 figures. This version corresponds to the version that is to appear in the Journal of Functional Analysis

We consider hypoelliptic equations of kinetic Fokker-Planck type, also known as Kolmogorov or ultraparabolic equations, with rough coefficients in the drift-diffusion operator. We give novel short quantitative proofs of the De Giorgi intermediate-value Lemma as well as weak Harnack and Harnack inequalities. This implies H{\"o}lder continuity with quantitative estimates. The paper is self-contained.

This paper gives the first affirmative answer to the question of the global existence of Boltzmann equations without angular cutoff in the $L^\infty$-setting. In particular, we show that when the initial data is close to equilibrium and the perturbation is small in $L^2 \cap L^\infty$ with a polynomial decay tail, the Boltzmann equation has a global solution in the weighted $L^2\cap L^\infty$-space. In order to overcome the difficulties arising from the singular cross-section and the low regularity, a De Giorgi type argument is crafted in the kinetic context with the help of the averaging lemma. More specifically, we use a strong averaging lemma to obtain suitable $L^p$-estimates for level-set functions. These estimates are crucial for constructing an appropriate energy functional to carry out the De Giorgi argument. Similar as in \cite{AMSY}, we extend local solutions to global ones by using the spectral gap of the linearised Boltzmann operator. The convergence to the equilibrium state is then obtained as a byproduct with relaxations shown in both $L^2$ and $L^\infty$-spaces., Comment: Removed the uniqueness statement from the previous version. The main body of the work is little changed since it is for the existence

We make several remarks concerning properties of functions in parabolic De Giorgi classes of order $p$. There are new perspectives including a novel mechanism of propagating positivity in measure, the reservation of membership under convex composition, and a logarithmic type estimate. Based on them, we are able to give new proofs of known properties. In particular, we prove local boundedness and local H\"older continuity of these functions via Moser's ideas, thus avoiding De Giorgi's heavy machinery. We also seize this opportunity to give a transparent proof of a weak Harnack inequality for non-negative members of some super-class of De Giorgi, without any covering argument.