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Despite significant recent advances in the regularity theory for obstacle problems with integro-differential operators, some fundamental questions remained open. On the one hand, there was a lack of understanding of parabolic problems with critical scaling, such as the obstacle problem for $\partial_t+\sqrt{-\Delta}$. No regularity result for free boundaries was known for parabolic problems with such scaling. On the other hand, optimal regularity estimates for solutions (to both parabolic and elliptic problems) relied strongly on monotonicity formulas and, therefore, were known only in some specific cases. In this paper, we present a novel and unified approach to answer these open questions and, at the same time, to treat very general operators, recovering as particular cases most previously known regularity results on nonlocal obstacle problems.

We introduce nonlocal minimal surfaces on closed manifolds and establish a far-reaching Yau-type result: in every closed, $n$-dimensional Riemannian manifold we construct infinitely many nonlocal $s$-minimal surfaces. We prove that, when $s\in (0,1)$ is sufficiently close to $1$, the constructed surfaces are smooth for $n=3$ and $n=4$, while for $n\ge 5$ they are smooth outside of a closed set of dimension $n-5$. Moreover, we prove surprisingly strong regularity and rigidity properties of finite Morse index $s$-minimal surfaces such as a "finite Morse index Bernstein-type result". These properties make nonlocal minimal surfaces ideal objects on which to apply min-max variational methods.

arXiv

Este livro suscita-nos, desde logo, a interroga����o sobre o aparente ox��moro que constitui o seu t��tulo: como �� que um museu ��� que �� por defini����o um espa��o fechado, interior, limitado ��� o pode ser da paisagem - que �� de sua natureza aberta, exterior e mais ou menos ilimitada? Ao que acresce uma outra raz��o para esta aparente antinomia entre museu e paisagem: o museu tem a ver com a hist��ria, com o tempo, com o passado - mesmo quando nele se figura um presente, ele �� figurado como j�� tendo sido; a paisagem tem a ver com a geografia, o espa��o, o presente - mesmo quando ela tem um passado, e tem-no sempre, o que n��s vemos nela �� sempre o presente., Comunica����o P��blica, Vol. 16 N.�� 31 (2021): N��mero com dossi�� tem��tico

Fusion nuclear energy is one of key energy resource towards switch to a more sustainable energy generation. Even though there is a deep understanding of the physics involved, there are still engineering challenges and technology limits that are needed to be broken to access this resource. One of the drawbacks are the cooling of the reactor and the tritium breeding. Most standardized layouts use what is so-called a breeding blanket (BB). The BB presents three main functions: shield the super-conductive magnets and the vacuum vessel from the reactor neutron flux, extract the heat which will be used to produce useful energy, and help to breed tritium (from the thermal neutrons, fast neutrons and the lithium) to be further used as fuel. The BB may embed internal channels to allow a breeder-coolant fluid to circulate along them. One of the current proposals uses two coolants, where the first one is based on a liquid metal alloy of Lithium and Lead, and the second one is helium. This design is also known as the Dual-Coolant Lithium Lead (DCLL). The behaviour of this coolant is critical concerning the heat transfer and tritium transport. These conditions are used to determine the temperature and tritium distribution inside the channel. Temperature indicates whether a material can withstand the imposed conditions as a structural component and the tritium distribution specifies if there might be an undesirable tritium accumulation inside the channel, since tritium is needed to be gathered to sustain the reactor fuel cycle. In order to understand the behaviour of the flow with the extreme conditions it is surrounded by, intense magnetic field and mixed convection (natural due to the high temperature gradient and forced). The physics the flows can be studied by magnetohydrodynamics (MHD) model using computational fluid dynamics (CFD) methods. Such models, require high computational resources to be resolved which make them time and economically expensive. The aim of the current project is contributing on the research and the development of analysis tools to improve the understanding of the technology and cheapen such development. Therefore, after working on this project, a detecting instabilities algorithm combination of two methods based on the Discrete Fourier Transform have been developed and validated. The results are promising because now there is a new tool ready to be fully tested and provide a new way of analysing large sets of simulations, specially when there is no time to manually examine one case at a time. Thus, it is expected to properly fit in the work chain of the study of the parametrization based on Re, Gr and Ha of the MHD-LM channel flow behaviour.

We study critical points of a one-parameter family of functionals arising in combustion models. The problems we consider converge, for infinitesimal values of the parameter, to Bernoulli's free boundary problem, also known as one-phase problem. We prove a C1,α estimates for the "interfaces" (level sets separating the burnt and unburnt regions). As a byproduct, we obtain the one-dimensional symmetry of minimizers in the whole ℝN, for N ≤ 4, answering positively a conjecture of Fernández-Real and Ros-Oton. Our results are to Bernoulli's free boundary problem what Savin's results for the Allen-Cahn equation are to minimal surfaces.

arXiv

In this paper we analyze the singular set in the Stefan problem and prove the following results: - The singular set has parabolic Hausdorff dimension at most $n-1$. - The solution admits a $C^\infty$-expansion at all singular points, up to a set of parabolic Hausdorff dimension at most $n-2$. - In $\mathbb R^3$, the free boundary is smooth for almost every time $t$, and the set of singular times $\mathcal S\subset \mathbb R$ has Hausdorff dimension at most $1/2$. These results provide us with a refined understanding of the Stefan problem's singularities and answer some long-standing open questions in the field.

We study stable solutions to the fractional Allen-Cahn equation \linebreak $(-\Delta)^{s/2} u = u-u^3$, $|u

For the thin obstacle problem in $\mathbb{R}^n$, $n\geq 2$, we prove that at all free boundary points, with the exception of a $(n-3)$-dimensional set, the solution differs from its blow-up by higher order corrections. This expansion entails a $C^{1,1}$-type free boundary regularity result, up to a codimension 3 set.

Uma breve panorâmica sobre a evolução da administração municipal eborense Se no capítulo anterior atendemos aos contextos, no presente iremos debruçar-nos sobre a estrutura do poder municipal e das suas dinâmicas. Não sendo este um estudo sobre a administração municipal, em si mesma, não faremos uma abordagem exaustiva a este nível. Centrar-nos-emos, apenas, nos principais cargos concelhios, a partir dos quais se geriam os destinos municipais, que foram aqueles que os nossos homens desempenha...

Procuraremos, neste capítulo, traçar um quadro genérico do período sobre o qual nos debruçamos, visto naturalmente a partir de Évora, numa análise que se revela indispensável para que possamos entender algumas das questões que colocámos em torno da administração municipal e dos homens que dirigiram os destinos da cidade. Muito embora o período final da nossa cronologia seja já de estabilidade, uma boa parte do tempo que a atravessa foi de profunda crise, com causas de natureza diversa. As gue...

O grupo da governança : critérios de elegibilidade e delimitação social Antes de identificarmos o conjunto de homens que exerceram o poder e analisarmos as suas carreiras, importa apresentar os critérios de elegibilidade para o exercício dos cargos, já que nem todos estavam habilitados a exercer funções na administração municipal. Como se sabe, o acesso aos cargos concelhios era limitado, existiam critérios e condições que a priori impediam o seu desempenho à grande maioria da população urb...

Governar a cidade e servir o rei é um estudo sobre o grupo dirigente que controlou o poder municipal, em Évora num período de cerca de 70 anos, correspondente aos reinados de D. Fernando e D. João I, passando pela crise dinástica de 1383-1385. Trata-se de uma análise dinâmica que levou, naturalmente, em consideração um contexto muito especial, vivido de forma intensa numa cidade que teve, como se sabe, um papel fulcral em todo o processo que conduziu o Mestre de Avis à coroa. Esse contexto, que o estudo enfatiza, não foi indiferente para os rumos da governação municipal, nem para os destinos daqueles que dirigiam a cidade, que, na maior parte dos casos, estiveram de forma desassombrada ao lado do mestre, não lhe negando o seu apoio militar e financeiro. Um tal posicionamento não só permitiu a esse grupo de famílias ligadas à governação reforçar a sua capacidade de controlo sobre o poder municipal, como acabou também por se constituir como a pedra de toque da sua ascenção social. O protagonismo assumido, nesses anos de fogo, à frente de uma cidade que conheceu, neste transcurso temporal, uma forte elevação no quadro político do reino, e que fez dela uma das principais cidades cortesãs, constituiu um impulso para esse grupo de famílias, em que se destacam os Lobo, os d'Arca, os Fuseiro, os Oliveira, os Façanha ou os Arnalho, permitindo-lhes iniciar ou acelerar processos de mobilidade social ascendente que, em alguns casos, os colocou no seio dos grupos nobiliárquicos. Neste sentido, o trabalho que agora chega a um público mais alargado é não só um estudo sobre o controlo do poder, mas também uma investigação sobre os mecanismos e as estratégias de ascenção social de homens, com origens sociais mais ou menos obscuras, para quem o domínio da governação municipal constituiu um trampolim de mobilidade que os aproximou dos grupos nobiliárquicos que constituíam o seu horizonte social.

Iniciaremos este capítulo com uma caraterização social do grupo dirigente, aprofundando um pouco mais o quadro genérico, que traçámos anteriormente a esse nível. Como veremos, apesar das restrições que limitavam o acesso ao poder municipal e da sua elitização, o grupo governativo estava longe de ser homogéneo sob o ponto de vista social. No seu interior, existiam diferentes níveis de poder, de riqueza e distintos estatutos, que se refletiam no próprio perfil das carreiras concelhias, como vim...

Os grupos dirigentes concelhios têm vindo a merecer, nas últimas décadas, uma redobrada atenção por parte dos medievalistas portugueses, assistindo-se a um inegável avanço relativamente a uma matéria pouco abordada pela historiografia tradicional. A atenção prestada à dimensão sociológica do poder concelhio integra-se num movimento mais vasto de renovação da historiografia medieval portuguesa, que se verificou a partir das décadas de oitenta e noventa do século passado, consubstanciado na abe...

Olhando globalmente para a administração municipal eborense, ao longo do período cronológico em estudo, um dos traços que fica evidente é a existência de uma linha de continuidade. Continuidade que os acontecimentos políticos e militares que marcaram fortemente este período e a subida de uma nova dinastia ao poder não interrompeu, não se podendo falar, propiamente, de ruturas. Não significa isso que, ao longo dos quase setenta anos abrangidos pelo nosso estudo, não se tivessem verificado muda...

We study the Plateau problem with a lower dimensional obstacle in $\mathbb{R}^n$. Intuitively, in $\mathbb{R}^3$ this corresponds to a soap film (spanning a given contour) that is pushed from below by a "vertical" 2D half-space (or some smooth deformation of it). We establish almost optimal $C^{1,1/2-}$ estimates for the solutions near points on the free boundary of the contact set, in any dimension $n\ge 2$. The $C^{1,1/2-}$ estimates follow from an $\varepsilon$-regularity result for minimal surfaces with thin obstacles in the spirit of the De Giorgi's improvement of flatness. To prove it, we follow Savin's small perturbations method. A nontrivial difficulty in using Savin's approach for minimal surfaces with thin obstacles is that near a typical contact point the solution consists of two smooth surfaces that intersect transversally, and hence it is not very flat at small scales. Via a new "dichotomy approach" based on barrier arguments we are able to overcome this difficulty and prove the desired result.

A Biblioteca Pública de Évora possui, como se sabe, um dos mais ricos espólios documentais do país, com fundos riquíssimos e em alguns casos únicos, constituindo uma instituição de referência para muitas áreas de investigação, desde a História à Antropologia, à Cartografia, à Literatura ou à Cultura Portuguesas, entre outras. É uma das mais antigas de Portugal. Foi fundada no ano de 1805, por um dos homens que simbolizam em pleno o ideário das luzes no nosso país, frei Manuel do Cenáculo, Arc...

A tese tem como fulcro o estudo do grupo ligado à administração municipal da cidade de Évora, no período compreendido entre 1367-1433. Procurou-se caraterizar socialmente esse grupo e analisar as suas estratégias de manutenção e de reprodução do poder, avaliando-se, em paralelo, o peso e a importância que o controlo dos cargos municipais assumia nas suas lógicas de afirmação, no contexto da cronologia escolhida. O trabalho em causa pretende, assim, ser um contributo para o aprofundamento do t...

We prove that half spaces are the only stable nonlocal $s$-minimal cones in $\mathbb{R}^3$, for $s\in(0,1)$ sufficiently close to $1$. This is the first classification result of stable $s$-minimal cones in dimension higher than two. Its proof can not rely on a compactness argument perturbing from $s=1$. In fact, our proof gives a quantifiable value for the required closeness of $s$ to $1$. We use the geometric formula for the second variation of the fractional $s$-perimeter, which involves a squared nonlocal second fundamental form, as well as the recent BV estimates for stable nonlocal minimal sets.

We establish sharp boundary regularity estimates in $C^1$ and $C^{1,\alpha}$ domains for nonlocal problems of the form $Lu=f$ in $\Omega$, $u=0$ in $\Omega^c$. Here, $L$ is a nonlocal elliptic operator of order $2s$, with $s\in(0,1)$. First, in $C^{1,\alpha}$ domains we show that all solutions $u$ are $C^s$ up to the boundary and that $u/d^s\in C^\alpha(\bar\Omega)$, where $d$ is the distance to $\partial\Omega$. In $C^1$ domains, solutions are in general not comparable to $d^s$, and we prove a boundary Harnack principle in such domains. Namely, we show that if $u_1$ and $u_2$ are positive solutions, then $u_1/u_2$ is bounded and H\"older continuous up to the boundary. Finally, we establish analogous results for nonlocal equations with bounded measurable coefficients in non-divergence form. All these regularity results will be essential tools in a forthcoming work on free boundary problems for nonlocal elliptic operators \cite{CRS-obstacle}.

We establish Pohozaev identities and integration by parts type formulas for anisotropic integro-differential operators of order $2s$, with $s\in(0,1)$. These identities involve local boundary terms, in which the quantity $u/d^s|_{\partial\Omega}$ plays the role that $\partial u/\partial\nu$ plays in the second order case. Here, $u$ is any solution to $Lu=f(x,u)$ in $\Omega$, with $u=0$ in $\mathbb R^n\setminus\Omega$, and $d$ is the distance to $\partial\Omega$.

We establish $C^{\sigma+\alpha}$ interior estimates for concave nonlocal fully nonlinear equations of order $\sigma\in(0,2)$ with rough kernels. Namely, we prove that if $u\in C^{\alpha}(\mathbb R^n)$ solves in $B_1$ a concave translation invariant equation with kernels in $\mathcal L_0(\sigma)$, then $u$ belongs to $C^{\sigma+\alpha}(\overline{ B_{1/2}})$, with an estimate. More generally, our results allow the equation to depend on $x$ in a $C^\alpha$ fashion. Our method of proof combines a Liouville theorem and a blow-up (compactness) procedure. Due to its flexibility, the same method can be useful in different regularity proofs for nonlocal equations.

We establish $C^{��+��}$ interior estimates for concave nonlocal fully nonlinear equations of order $��\in(0,2)$ with rough kernels. Namely, we prove that if $u\in C^��(\mathbb R^n)$ solves in $B_1$ a concave translation invariant equation with kernels in $\mathcal L_0(��)$, then $u$ belongs to $C^{��+��}(\overline{ B_{1/2}})$, with an estimate. More generally, our results allow the equation to depend on $x$ in a $C^��$ fashion. Our method of proof combines a Liouville theorem and a blow-up (compactness) procedure. Due to its flexibility, the same method can be useful in different regularity proofs for nonlocal equations.

Despite it being a debated issue in recent decades, there is in fact no consensus regarding the potentialities of the Internet for the strengthening of citizens’ political participation and – by extension – of democracy itself. Nonetheless, this “new medium”, and, in particular, its Web 2.0 version has become the cornerstone of communication strategies for both political parties and their candidates. Several questions emerge from the debate about the impact of the internet on political participation. First of all, the very concept of political participa- tion. Despite its importance in the theory of democracy, this concept is not always easy to define with rigor particularly when it comes to the form and boundaries of such participation. Actually, the ultimate issue is whether we are currently witnessing a “crisis of participation” or quite the contrary, the rise of alternative forms thereof. Secondly, the debate about the so-called “crisis of democracy” is also central when thinking about political participa- tion in our times. Democratic societies are experiencing a democratic deficit, to a large extend a crisis of confidence in traditional political parties, by which citizens feel misrepresented and hence alienated from righteous participation in their political destiny.

Podeu consultar el llibre complet a: http://hdl.handle.net/2445/32166, Precession electron diffraction (PED) is a hollow cone non-stationary illumination technique for electron diffraction pattern collection under quasikinematical conditions (as in X-ray Diffraction), which enables “ab-initio” solving of crystalline structures of nanocrystals. The PED technique is recently used in TEM instruments of voltages 100 to 300 kV to turn them into true electron iffractometers, thus enabling electron crystallography. The PED technique, when combined with fast electron diffraction acquisition and pattern matching software techniques, may also be used for the high magnification ultra-fast mapping of variable crystal orientations and phases, similarly to what is achieved with the Electron Backscatter Diffraction (EBSD) technique in Scanning Electron Microscopes (SEM) at lower magnifications and longer acquisition times.

El principal objectiu del meu projecte final de carrera és realitzar una aplicació web eficient i usable utilitzant el servidor de captures que utilitzava l'aplicació d'escriptori que em va proporcionar l'Isaac Gelado utilitzant tecnologies “open source”. El primer objectiu és que l'aplicació mostri captures per pantalla. Dins d'aquest apartat podríem definir objectius menys importants però no per això menys necessaris. Guardar les captures, veure els diferents sniffers que es troben a la xarxa, poder filtrar captures per ip, port i protocol, reconfigurar el filtre de captures són objectius que s'hauran d'anar assolint un cop s'hagi dut a terme el primer objectiu esmentat anteriorment. Un dels objectius més importants és que l'aplicació sigui eficient i ràpida. Que el codi que la forma sigui robust i les tecnologies utilitzades també. Caldrà, per tant, raonar molt bé alhora de prendre decisions de disseny de l'aplicació. L'aplicació haurà de ser vistosa i entenedora per tant els estils i les decisions estructurals també hauran d'ésser meditades amb deteniment.

El principal objectiu del meu projecte final de carrera és realitzar una aplicació web eficient i usable utilitzant el servidor de captures que utilitzava l'aplicació d'escriptori que em va proporcionar l'Isaac Gelado utilitzant tecnologies “open source”. El primer objectiu és que l'aplicació mostri captures per pantalla. Dins d'aquest apartat podríem definir objectius menys importants però no per això menys necessaris. Guardar les captures, veure els diferents sniffers que es troben a la xarxa, poder filtrar captures per ip, port i protocol, reconfigurar el filtre de captures són objectius que s'hauran d'anar assolint un cop s'hagi dut a terme el primer objectiu esmentat anteriorment. Un dels objectius més importants és que l'aplicació sigui eficient i ràpida. Que el codi que la forma sigui robust i les tecnologies utilitzades també. Caldrà, per tant, raonar molt bé alhora de prendre decisions de disseny de l'aplicació. L'aplicació haurà de ser vistosa i entenedora per tant els estils i les decisions estructurals també hauran d'ésser meditades amb deteniment.

Caracterização do grupo que governava Évora no séc. XV do ponto de vista social e político.

O artigo debruça-se sobre fundação do mosteiro Jerónimo de Santa Maria do Espinheiro, no espaço periurbano de Évora, por meados do século XV

o trabalho debruça-se sobre a organização do espaço em torno de Évora, uma das mais importantes cidades medievais portuguesas

Submitted by Bruno Leal (contemporanea.cehr@ft.lisboa.ucp.pt) on 2011-06-22T11:22:07Z No. of bitstreams: 1 LS_S2_17_JoaquimBSerra.pdf: 185174 bytes, checksum: 7748bae70460b4046bf6f9c62710f372 (MD5) Approved for entry into archive by Margarida Carvalho(mvc@lisboa.ucp.pt) on 2011-06-22T18:31:52Z (GMT) No. of bitstreams: 1 LS_S2_17_JoaquimBSerra.pdf: 185174 bytes, checksum: 7748bae70460b4046bf6f9c62710f372 (MD5) Made available in DSpace on 2011-06-22T18:31:52Z (GMT). No. of bitstreams: 1 LS_S2_17_JoaquimBSerra.pdf: 185174 bytes, checksum: 7748bae70460b4046bf6f9c62710f372 (MD5) Previous issue date: 2005

El principal objectiu del meu projecte final de carrera és realitzar una aplicació web eficient i usable utilitzant el servidor de captures que utilitzava l'aplicació d'escriptori que em va proporcionar l'Isaac Gelado utilitzant tecnologies “open source”. El primer objectiu és que l'aplicació mostri captures per pantalla. Dins d'aquest apartat podríem definir objectius menys importants però no per això menys necessaris. Guardar les captures, veure els diferents sniffers que es troben a la xarxa, poder filtrar captures per ip, port i protocol, reconfigurar el filtre de captures són objectius que s'hauran d'anar assolint un cop s'hagi dut a terme el primer objectiu esmentat anteriorment. Un dels objectius més importants és que l'aplicació sigui eficient i ràpida. Que el codi que la forma sigui robust i les tecnologies utilitzades també. Caldrà, per tant, raonar molt bé alhora de prendre decisions de disseny de l'aplicació. L'aplicació haurà de ser vistosa i entenedora per tant els estils i les decisions estructurals també hauran d'ésser meditades amb deteniment.

“The final publication is available at Springer via http://dx.doi.org/10.1007/s10231-016-0632-1" We establish sharp boundary regularity estimates in C1 and C1,a domains for nonlocal problems of the form Lu=f in O, u=0 in Oc. Here, L is a nonlocal elliptic operator of order 2s, with s¿(0,1). First, in C1,a domains we show that all solutions u are Cs up to the boundary and that u/ds¿Ca(O¯¯¯¯), where d is the distance to ¿O. In C1 domains, solutions are in general not comparable to ds, and we prove a boundary Harnack principle in such domains. Namely, we show that if u1 and u2 are positive solutions, then u1/u2 is bounded and Hölder continuous up to the boundary. Finally, we establish analogous results for nonlocal equations with bounded measurable coefficients in nondivergence form. All these regularity results will be essential tools in a forthcoming work on free boundary problems for nonlocal elliptic operators (Caffarelli et al., in Invent Math, to appear).

Given an arbitrary convex cone of Rn, we find a geometric class of homogeneous weights for which balls centered at the origin and intersected with the cone are minimizers of the weighted isoperimetric problem in the convex cone. This leads to isoperimetric inequalities with the optimal constant that were unknown even for a sector of the plane. Our result applies to all nonnegative homogeneous weights in Rn satisfying a concavity condition in the cone. The condition is equivalent to a natural curvature-dimension bound and also to the nonnegativeness of a Bakry-Emery Ricci tensor. Even that our weights are nonradial, still balls are minimizers of the weighted isoperimetric problem. A particular important case is that of monomial weights. Our proof uses the ABP method applied to an appropriate linear Neumann problem. We also study the anisotropic isoperimetric problem in convex cones for the same class of weights. We prove that the Wulff shape (intersected with the cone) minimizes the anisotropic weighted perimeter under the weighted volume constraint. As a particular case of our results, we give new proofs of two classical results: the Wulff inequality and the isoperimetric inequality in convex cones of Lions and Pacella

We study critical points of a one-parameter family of functionals arising in combustion models. The problems we consider converge, for infinitesimal values of the parameter, to Bernoulli's free boundary problem, also known as one-phase problem. We prove a C1,α estimates for the “interfaces” (level sets separating the burnt and unburnt regions). As a byproduct, we obtain the one-dimensional symmetry of minimizers in the whole RN, for N≤4, answering positively a conjecture of Fernández-Real and Ros-Oton. Our results are to Bernoulli's free boundary problem what Savin's results for the Allen-Cahn equation are to minimal surfaces., Advances in Mathematics, 403, ISSN:0001-8708, ISSN:1090-2082

[eng] The application of electron diffraction to crystallographically characterize all kinds of materials has experienced new developments that have attracted some attention in recent years. A large number of structural analyses from different compounds have been already carried out with the help of 3D electron diffraction data that were not possible with the available X-ray methods. The use of a transmission electron microscope as an electron nano-diffractometer has proved to be advantageous when diffraction data from single nanocrystals are required, for instance in phase mixtures. In this way, the individual phases do not have to be purely synthesized, which always involves the risk of structural changes. The work presented here includes the development and implementation of a novel and universal routine for the accurate and reliable acquisition of electron diffraction data. It contains the alignment of the transmission electron microscope in order to set a quasi-parallel and precessed beam suitable for the acquisition of this kind of data, and the description and comparison of the developed technique to the other methods already available. This novel acquisition method has been called fast and automated diffraction tomography (Fast-ADT) and it is based on two consecutive tilt scans of the goniometric stage of the microscope; one to image the crystal and generate a crystal tracking file, and a second one to acquire the diffraction patterns while the beam is automatically shifted to follow the crystal at the different tilt angles. Such technique has been implemented in different experimental setups to test different detectors as well as different samplings of the diffraction space to point out their advantages and hindrances. The potential of this new data collection strategy to solve various crystallographic problems is illustrated using three known materials: barite, an inorganic salt stable under the electron beam used to check the level of accuracy that can be obtained from electron diffraction, RUB-5, a layered silicate with disordered features to show the advantage of high frame-rate cameras, and the pi-ferrosilicide, a crystalline phase from a commercial alloy measured to demonstrate how electron diffraction can reliably determine absolute structures. In addition, two unknown crystal structures from commercial products are fully determined and refined. The first one is the dehydrated DRED1 molecular crystal, an inorganic dye used in textile applications, which has been solved ab initio revealing the 46 non-hydrogen atoms of the two independent molecules in P-1. Furthermore, the structure refinement based on intensities calculated according to the dynamical diffraction theory has proved the position of some of the hydrogens, confirming the square-like H-bond network initially found from a Rietveld refinement based on a X-ray powder diffractogram. Finally, the incommensurate modulated structure of one of the polymorphs of dicalcium silicate (belite), a major constituent of cement, has been comprehensively characterized only by means of electron diffraction. Such crystal structure has been known since 1971 but the observed modulation has never been fully studied. Here, the Fast-ADT technique is presented as a robust acquisition method to systematically identify and properly characterize different crystallographic phases and features of powder mixtures. In this particular case, the precise knowledge of the different crystal structures in cement clinkers, such as the alpha’H-C2S, enables the exact phase analysis of these industrial phase mixtures directly from the manufacturing process, and paves the way for its CO2 emissions reduction that, at the moment, adds up to 5% of the global anthropogenic emission. In this context, a novel technique for the collection of 3D electron diffraction datasets has been successfully developed and implemented to increase the range of reliable available tools for the characterization of atomic structures from different materials., [cat] L’aplicació de la difracció d’electrons per caracteritzar cristal·logràficament tot tipus de materials ha experimentat nous desenvolupaments que han cridat certa atenció durant aquests últims anys. Un gran nombre d’anàlisis estructurals en diferents compostos ja s’han dut a terme amb l’ajut de dades tridimensionals de difracció d’electrons que no eren possibles amb els habituals mètodes de raigs X. L’ús d'un microscopi electrònic de transmissió com a nano-difractòmetre d’electrons ha demostrat ser molt més beneficiós quan es requereixen dades de difracció de nanocristalls individuals, per exemple en mescles de fases cristal·logràfiques. D’aquesta manera, les fases individuals no s’han de sintetitzar en estat pur, fet que sempre comporta el risc de canvis estructurals. El treball presentat aquí inclou el desenvolupament i implementació d’una nova rutina universal per l’adquisició precisa i fiable de dades de difracció d’electrons. El potencial d’aquesta nova estratègia de recopilació de dades per resoldre diversos problemes cristal·logràfics s’il·lustra mitjançant tres materials coneguts. A més a més, dues estructures cristal·lines desconegudes de productes comercials han estat determinades i refinades completament; un colorant orgànic de baixa simetria i una estructura modulada incommensurada d’un component principal del ciment. En particular, el coneixement precís de les diferents estructures cristal·lines dels clinkers de ciment, com ara la alpha’H-C2S, permet l’anàlisi exacta de les fases d'aquestes mescles industrials extretes directament del procés de fabricació, i facilita el seu estudi per reduir les emissions de CO2.

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We establish quantitative properties of minimizers and stable sets for nonlocal interaction functionals, including the $s$-fractional perimeter as a particular case. ¶ On the one hand, we establish universal $BV$-estimates in every dimension $n \geqslant 2$ for stable sets. Namely, we prove that any stable set in $B_1$ has finite classical perimeter in $B_{1/2}$, with a universal bound. This nonlocal result is new even in the case of $s$-perimeters and its local counterpart (for classical stable minimal surfaces) was known only for simply connected two-dimensional surfaces immersed in $\mathbb{R}^3$. ¶ On the other hand, we prove quantitative flatness estimates for minimizers and stable sets in low dimensions $n = 2, 3$. More precisely, we show that a stable set in $B_R$, with $R$ large, is very close in measure to being a half space in $B_1$ – with a quantitative estimate on the measure of the symmetric difference. As a byproduct, we obtain new classification results for stable sets in the whole plane.

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