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This study uses luminescence and 14C accelerator mass spectrometry procedures to date relevant glaciofluvial and glacial deposits from the south-central and southeastern Pyrenees (Andorra–France–Spain). We distinguish two types of end-moraine complexes: (1) those in which at least a far-flung moraine exists beyond a frequently nested end-moraine complex (the most common) and (2) those in which a close-nested end moraine encompasses at least two glacial cycles. Both types formed within six distinctive glacial intervals: (1) A penultimate glacial cycle during Marine Oxygen Isotope Stage (MIS) 6 and older glaciofluvial terraces occurred beyond the range of the luminescence dating method. (2) An early glacial advance in MIS 5d (~97 −15/+19 ka) was followed by glacial retreat during MIS 5c (< 91 ± 9 ka). (3) The last maximum ice extent (LMIE) was in early MIS 4 (~74 ± 4.5 ka). (4) Unexpectedly, glaciers thinned during the second half of MIS 3 (~39 −6/+11 ka). (5) During the MIS 3–2 transition, glaciers subsequently fluctuated behind the LMIE limits. (6) The global last glacial maximum (LGM) started as early as ~26.6 ± 0.365 ka b2k, and the corresponding end moraines were built behind the LMIE limits or merged with it, forming close-nested moraines.

The Roca San Miguel (RSM) archaeological site was occupied during Mousterian times. Here we present a geoarchaeological and paleoenvironmental reconstruction of the site. Five stratigraphic units (A to E) formed by different archaeological levels are identified. Three optically stimulated luminescence (OSL) ages show that Unit A dates to between 169.6 ± 9.1 and 151.9 ± 11.1 ka, during the penultimate glacial period (PGP), and contains numerous signs of recurring hearths. Unit B is unexcavated. Unit C dates to between 118.9 ± 11.5 and 103.4 ± 6.9 ka (late Eemian–marine isotope stage (MIS) 5d) and shows an abundance of lithic remains as well as some faunal elements. Unit C is covered by Unit D, which incorporates materials moved downslope, and is dated at 81.2 ± 4.7 ka. These OSL ages concur with U/Th ages (129.3 ± 1.5 and 123.6 ± 0.6 ka) derived from a flowstone covered by both -C and Dpost-flowstone units. Finally, Unit E covers the archaeological site, which was partially eroded during MIS2. The robust and well-constrained chronology of the RSM site and surroundings enables the establishment of its evolutionary model from the PGP to the last glacial cycle. The RSM site is the oldest Neanderthal occupation accurately dated in the Pre-Pyrenean region., This research was supported by the Gaps and Sites (HAR2017-85023-P) and Gaps and Dates (PID2020-116598GB-I00) projects, funded by the MINECO/MICIN-AEI/FEDER, Spain/EU; and by the Research Group P3A from the Aragón Government (H14_20R).

Trip patterns in terms of origin-to-destination (OD) traffic flows are a key input to traffic assignment models, namely to dynamic traffic assignment models. The trip patterns must also be dynamic – or at least time discretized – to properly approximate the time variability of the demand. However, OD matrices are not yet observable; in the best case, the measurements from Information and Communications Technologies (ICT) such as GPS vehicle tracking or mobile phone call detail records (CDR) can be used to collect samples that must then be used to infer or estimate the whole population. In order to perform this estimation, indirect processes that are usually based on mathematical models that take as input traffic counts are most commonly used. The problem of OD matrix estimation has been interesting for both researchers and practitioners for decades, (Van Zuylen and Willumsen, 1980). Furthermore, the aim of capturing the congestion effects of the changing OD flows during the OD estimation process conducted the research to an appealing mathematical formulation as a bi-level optimization problem. Moreover, in the recent years, the dynamic traffic assignments arise in order to capture the evolution of the traffic system according its dynamicity, leading to the dynamic OD matrix estimation problem. Peer Reviewed Objectius de Desenvolupament Sostenible::11 - Ciutats i Comunitats Sostenibles::11.3 - Per a 2030, augmentar la urbanització inclusiva i sostenible, així com la capacitat de planificar i gestionar de manera participativa, integrada i sostenible els assentaments humans a tots els països Objectius de Desenvolupament Sostenible::11 - Ciutats i Comunitats Sostenibles

Since OD matrices are not directly observable, indirect procedures have been developed to estimate OD matrices from traffic data. Traffic management must move toward dynamic traffic assignment models because they capture congestion propagation effects. In this context, dynamic OD matrices are needed. This paper first explores the extension of a conventional bilevel analytical static method to the dynamic context, then analyses the solutions obtained regarding both convergence to measured traffic data and structural similarity to an a priori OD matrix. A simulation optimization technique, SPSA, is then proposed because its flexibility allows inclusion of traffic counts (as in Spiess method) and emerging ICT traffic measurements. The performances of these two types of algorithms are analyzed in detail, focussing on faults of classical convergence measures to obtain an estimated dynamic OD matrix structurally similarity to the a priori OD matrix and pros and cons of variants of selected methods.

We prove new boundary Harnack inequalities in Lipschitz domains for equations with a right hand side. Our main result applies to non-divergence form operators with bounded measurable coefficients and to divergence form operators with continuous coefficients, whereas the right hand side is in L q with q > n . Our approach is based on the scaling and comparison arguments of [13] , and we show that all our assumptions are sharp. As a consequence of our results, we deduce the C 1 , α regularity of the free boundary in the fully nonlinear obstacle problem and the fully nonlinear thin obstacle problem.

We study solutions to $Lu=f$ in $\Omega\subset\mathbb R^n$, being $L$ the generator of any, possibly non-symmetric, stable L\'evy process. On the one hand, we study the regularity of solutions to $Lu=f$ in $\Omega$, $u=0$ in $\Omega^c$, in $C^{1,\alpha}$ domains~$\Omega$. We show that solutions $u$ satisfy $u/d^\gamma\in C^{\varepsilon_\circ}\big(\overline\Omega\big)$, where $d$ is the distance to $\partial\Omega$, and $\gamma=\gamma(L,\nu)$ is an explicit exponent that depends on the Fourier symbol of operator $L$ and on the unit normal $\nu$ to the boundary $\partial\Omega$. On the other hand, we establish new integration by parts identities in half spaces for such operators. These new identities extend previous ones for the fractional Laplacian, but the non-symmetric setting presents some new interesting features. Finally, we generalize the integration by parts identities in half spaces to the case of bounded $C^{1,\alpha}$ domains. We do it via a new efficient approximation argument, which exploits the H\"older regularity of $u/d^\gamma$. This new approximation argument is interesting, we believe, even in the case of the fractional Laplacian.

Geological and geophysical studies in complex valley troughs provide a key record for the reconstruction of paleoenvironmental conditions during the Quaternary. Here we present a study of the sedimentary infill of the El Cervunal kame complex or El Cervunal trough (Sierra de Gredos, Iberian Central System) by means of a combined interpretation of near-surface geophysical techniques supported by geomorphological and borehole data. A set of 1D and 2D near-surface geophysical methods, including electrical (Vertical Electrical Sounding and 2D Electrical Resistivity Tomography), seismic (2D Seismic Tomography and 1D Refraction Microtremor) and Magnetic Resonance Sounding techniques, were used to test their applicability in providing better insight on the infill nature and geometry. Because of greater sensitivity and higher resolution and coverage achieved with resistivity methods, the electrical resistivity has proven to be the most informative physical parameter, while seismic and magnetic resonance methods were complementary. The sedimentary infill was classified into three geophysical units and five sub-units with their geological interpretation. Unit 1 consists of a postglacial sedimentary sequence and includes alluvial-plain and alluvial-fan deposits. Units 2 and 3 below the postglacial unit were interpreted as glacial sequences including kame (glacigenetic and fluvio-glacial) and morainic deposits, respectively. Subsoil information combined with the geomorphological data enabled the partial reconstruction of the map-view geometry of the morainic bodies at the El Cervunal trough. The results suggest a complex evolution of the study area where erosive-sedimentary processes dominate, but also structural factors should be considered. At least five well-differentiated stages can be established to explain the occurrence of the trough infill: pre-glacial, maximum glacial extension, trough obturation, glacial retreat and periglacial-postglacial., This work was mainly supported by the Spanish Ministry of Science and Innovation (Spanish initials: MINECO; project CGL2016-78380-P) and UCLM Research Groups (Project 2020-GRIN-28992). The authors acknowledge the help received from the Sierra de Gredos Regional Park and the Consejería de Medio Ambiente of the Junta de Castilla y León (JCyL). Additional support comes from the Program I+D CARESOIL (S2018/EMT-4317) of the Comunidad Autónoma of Madrid

Free boundary problems are those described by PDE that exhibit a priori unknown (free) interfaces or boundaries. Such type of problems appear in Physics, Geometry, Probability, Biology, or Finance, and the study of solutions and free boundaries uses methods from PDE, Calculus of Variations, and Geometric Measure Theory. The main mathematical challenge is to understand the regularity of free boundaries. The Stefan problem and the obstacle problem are the most classical and motivating examples in the study of free boundary problems. A milestone in this context is the classical work of Caffarelli, in which he established for the first time the regularity of free boundaries in the obstacle problem, outside a certain set of singular points. This is one of the main results for which he got the Wolf Prize in 2012 and the Shaw Prize in 2018.

In this paper we give a full classification of global solutions of the obstacle problem for the fractional Laplacian (including the thin obstacle problem) with compact coincidence set and at most polynomial growth in dimension N ≥ 3 . We do this in terms of a bijection onto a set of polynomials describing the asymptotics of the solution. Furthermore we prove that coincidence sets of global solutions that are compact are also convex if the solution has at most quadratic growth.

La cuenca lacustre de Banyoles-Besalú ha proporcionado un gran n-umero de yacimientos paleontológicos y arqueológicos en los que se encuentra bien representada una rica fauna de mamíferos que abarca un período que se extiende desde finales del Plio­ceno hasta el Cuaternario reciente. En la mayoría de los casos estos yacirnientos han sido ya estudiados y descritos, en especial los arqueológicos, conocidos desde finales del siglo XIX. En este trabajo se recogen las citas publicadas y se aportan nuevos datos como resultado de las revisiones que se estan efectuado en las tesis doctorates de dos de los autores (A. G. y J. M).

Se describe en detalle un esqueleto bastante completo de rinoceronte hallado en el yacimiento plioceno superior (Villafranquiense, MN 16-17) de Crespià (Cuenca de Banyoles-Besalu, Gerona, España). Se confirma su atribución preliminar a Stephanorhinus etruscus Falconer. Se establecen comparaciones con yacimientos comprendidos en el área de distribución bioestratigráfica de la especie y con algunos situados en los niveles inferiores del Pleistoceno medio donde aparece S. hundsheimensis Toula. El esqueleto corresponde a un individuo adulto, con una masa corporal estimada en 1200 kg. A nivel morfométrico, presenta claras semejanzas con otros restos españoles de la especie, como los de La Puebla de Valverde o El Rincón-1. Comparado con otros restos europeos, las dimensiones en general están por debajo de la media establecida para la especie y la serie premolar inferior es relativamente larga. Como detalles morfologicos característicos, destaca la sínfisis mandibular acortada y las ramas ascendentes relativamente bajas con respecto a algunos ejemplares coetáneos; dentición superior con gancho menos desarrollado que los ejemplares del Pleistoceno inferior; y huesos largos más gráciles que los de La Puebla de Valverde, aunque los metápodos mantienen epífisis relativamente anchas.

We study the obstacle problem for the fractional Laplacian with drift, $$\min \{(-\varDelta )^su + b \cdot \nabla u,\,u -\varphi \} = 0$$ in $$\mathbb {R}^n$$ , in the critical regime $$s = \frac{1}{2}$$ . Our main result establishes the $$C^{1,\alpha }$$ regularity of the free boundary around any regular point $$x_0$$ , with an expansion of the form where $$e \in \mathbb {S}^{n-1}$$ is the normal vector to the free boundary, $$\sigma >0$$ , and $$c_0> 0$$ . We also establish an analogous result for more general nonlocal operators of order 1. In this case, the exponent $$\tilde{\gamma }(x_0)$$ also depends on the operator.

We find and prove new Pohozaev identities and integration by parts type formulas for anisotropic integrodifferential operators of order 2s, with s¿(0,1). These identities involve local boundary terms, in which the quantity (Formula presented.) plays the role that ¿u/¿¿ plays in the second-order case. Here, u is any solution to Lu = f(x,u) in O, with u = 0 in RnO, and d is the distance to ¿O.

In this paper we prove the following long-standing conjecture: stable solutions to semilinear elliptic equations are bounded (and thus smooth) in dimension $n \leq 9$. This result, that was only known to be true for $n\leq4$, is optimal: $\log(1/|x|^2)$ is a $W^{1,2}$ singular stable solution for $n\geq10$. The proof of this conjecture is a consequence of a new universal estimate: we prove that, in dimension $n \leq 9$, stable solutions are bounded in terms only of their $L^1$ norm, independently of the nonlinearity. In addition, in every dimension we establish a higher integrability result for the gradient and optimal integrability results for the solution in Morrey spaces. As one can see by a series of classical examples, all our results are sharp. Furthermore, as a corollary we obtain that extremal solutions of Gelfand problems are $W^{1,2}$ in every dimension and they are smooth in dimension $n \leq 9$. This answers to two famous open problems posed by Brezis and Brezis-V\'azquez., Comment: To appear in Acta Mathematica

A new record from a long sediment core (S3) in Navamuño (1505 m asl, western Iberian Central System) provides the reconstruction of the vegetation history and environmental changes in the region between 15.6 and 10.6 ka cal BP, namely during the Late Glacial and the early Holocene, using a multiproxy analysis (pollen-based vegetation and climate reconstruction, sedimentary macrocharcoals, loss-on-ignition, magnetic susceptibility and X-ray fluorescence (XRF) measurements). The results are then compared with other sequences from the Iberian Central System and the whole Iberian Peninsula in order to better understand the past dynamics of the main forest constituents. The pollen record shows a shift from open pine forests ∼15.6–14.7 ka cal BP (Oldest Dryas) to mixed open pine-birch woodlands ∼14.7–14.0 ka cal BP (Bølling). Woodlands were succeeded by a steppe-like landscape until ∼13.4 ka cal BP (Older Dryas), which was replaced again by high-mountain pine forests and riparian woodlands ∼13.4–12.6 ka cal BP (Allerød). A great development of cold steppe grasslands linked to the decline of birch woodlands is documented ∼12.6–11.7 ka cal BP (Younger Dryas). The early Holocene (11.7–10.6 ka cal BP) was characterized by a progressive reforestation of the study area by pine and birch forests in the highlands and oak woods in the lowlands. Temperate tree taxa (Carpinus betulus, Castanea sativa, Corylus avellana, Fraxinus, Juglans, Tilia, and Ulmus) were also common but likely at lower elevations. Pollen of Fagus sylvatica was already recorded during the Late Glacial and the early Holocene. The marked increasing local fire activity during the warmer and wetter Allerød oscillation could be related to a rise in tree cover, supporting the climatically driven character of these fires. Nevertheless, the strong increase in fire activity during the Younger Dryas would probably be related to growing tree and shrub mortality, as well as to the wet/dry biphasic structure of this stadial. The standard “Modern Analogue Technique” has been also applied to the Navamuño sequence to provide quantitative climate estimations for the Late Glacial and the early Holocene periods. This record is one of the few continental archives that show the climatic trend between the Late Glacial and the early Holocene in central Iberia, agreeing with many other regional records from the Western Mediterranean.

In this note we study the boundary regularity of solutions to nonlocal Dirichlet problems of the form $Lu=0$ in $\Omega$, $u=g$ in $\mathbb R^N\setminus\Omega$, in non-smooth domains $\Omega$. When $g$ is smooth enough, then it is easy to transform this problem into an homogeneous Dirichlet problem with a bounded right hand side, for which the boundary regularity is well understood. Here, we study the case in which $g\in C^{0,\alpha}$, and establish the optimal H\"older regularity of $u$ up to the boundary. Our results extend previous results of Grubb for $C^\infty$ domains $\Omega$., Comment: 15 pages

We study the singular part of the free boundary in the obstacle problem for the fractional Laplacian, $\min \left\{(-\Delta)^s u, u-\varphi\right\}=0$ in $\mathbb{R}^n$, for general obstacles $\varphi$. Our main result establishes the complete structure and regularity of the singular set. To prove it, we construct new monotonicity formulas of Monneau-type that extend those in those of Garofalo-Petrosyan to all $s \in(0,1)$.

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 \(u/d^s\in C^\alpha (\overline{\Omega })\), 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 Holder 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).

We study the positivity and regularity of solutions to the fractional porous medium equations $u_t+(-\Delta)^su^m=0$ in $(0,\infty)\times\Omega$, for $m>1$ and $s\in (0,1)$ and with Dirichlet boundary data $u=0$ in $(0,\infty)\times({\mathbb R}^N\setminus\Omega)$, and nonnegative initial condition $u(0,\cdot)=u_0\geq0$. Our first result is a quantitative lower bound for solutions which holds for all positive times $t>0$. As a consequence, we find a global Harnack principle stating that for any $t>0$ solutions are comparable to $d^{s/m}$, where $d$ is the distance to $\partial\Omega$. This is in sharp contrast with the local case $s=1$, in which the equation has finite speed of propagation. After this, we study the regularity of solutions. We prove that solutions are classical in the interior ($C^\infty$ in $x$ and $C^{1,\alpha}$ in $t$) and establish a sharp $C^{s/m}_x$ regularity estimate up to the boundary. Our methods are quite general, and can be applied to a wider class of nonlocal parabolic equations of the form $u_t-\mathcal L F(u)=0$ in $\Omega$, both in bounded or unbounded domains.

© 2020. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/ There are presently several companies that provide processed or raw global positioning system (GPS) measurements generated by fleets of commercial vehicles, internet applications or automobile companies. The aim of this paper is to deepen our understanding of GPS data applicability in transportation modelling by providing systematic and quantified insights into the representativeness of collected data in describing individual’ route choices. Unfortunately, real data often contain noise, uncertainty, errors, redundancies or even irrelevant information. Useless models will be obtained when built over incorrect or incomplete data. This is why pre-processing is one of the most critical steps in data analysis. Yet, pre-processing has not been properly systematized, which is why this paper focuses on the necessary steps for pre-processing GPS tracking data together and puts forth a proposal for systematizing it. Furthermore, the aggregation level is at waypoint location for low latency GPS positions along the trip trajectory. Travel time reliability on OD paths is addressed, along with other OD path characteristics. Dense urban networks pose multiple possibilities for route choice, and data from new technologies offers an opportunity to understand route choice behaviour. This research was funded by TRA2016-76914-C3-1-P Spanish R+D Programs and by Secretaria d’Universitats-iRecerca-Generalitat de Catalunya- 2017-SGR- 1749 and Industrial PhD Program 2017 DI 041.

We investigate the regularity of the free boundary for the Signorini problem in $${\mathbb {R}}^{n+1}$$ R n + 1 . It is known that regular points are $$(n-1)$$ ( n - 1 ) -dimensional and $$C^\infty $$ C ∞ . However, even for $$C^\infty $$ C ∞ obstacles $$\varphi $$ φ , the set of non-regular (or degenerate) points could be very large—e.g. with infinite $${\mathcal {H}}^{n-1}$$ H n - 1 measure. The only two assumptions under which a nice structure result for degenerate points has been established are when $$\varphi $$ φ is analytic, and when $$\Delta \varphi < 0$$ Δ φ < 0 . However, even in these cases, the set of degenerate points is in general $$(n-1)$$ ( n - 1 ) -dimensional—as large as the set of regular points. In this work, we show for the first time that, “usually”, the set of degenerate points is small. Namely, we prove that, given any $$C^\infty $$ C ∞ obstacle, for almost every solution the non-regular part of the free boundary is at most $$(n-2)$$ ( n - 2 ) -dimensional. This is the first result in this direction for the Signorini problem. Furthermore, we prove analogous results for the obstacle problem for the fractional Laplacian $$(-\Delta )^s$$ ( - Δ ) s , and for the parabolic Signorini problem. In the parabolic Signorini problem, our main result establishes that the non-regular part of the free boundary is $$(n-1-\alpha _\circ )$$ ( n - 1 - α ∘ ) -dimensional for almost all times t, for some $$\alpha _\circ > 0$$ α ∘ > 0 . Finally, we construct some new examples of free boundaries with degenerate points.

OD matrices that describe mobility patterns provide major input to most transport analysis models. Since OD matrices are not yet directly observable, they are usually estimated indirectly. Data from new sources such as mobile phone records and GPS traces from mobile apps are emerging alternatives that allow for cheaper and timely estimates. However, they are still hindered by weaknesses that must be studied for use as input to transportation models. This paper presents a case study using mobility data from mobile phone records, with the goal of establishing a methodology for validating the obtained OD matrices and generating the appropriate input for traffic assignment models. Spatial and temporal consistency of OD data elaborated from mobile records has been proved to be useful for transportation modelling needs.

While optimization models play a key role in transportation analysis, the objective function to be optimized, however, cannot always be defined analytically. It is therefore necessary to resort to non-differentiable optimization methods that usually pivot on evaluating the objective function. Special cases of particular interest are those in Dynamic OD Estimation, which cannot evaluate the objective analytically and thus the formulation falls within the computational framework of Simulation-Optimization. SPSA is not limited to the inputs from conventional traffic counts and can be easily extended to account for the measurements of traffic variables supplied by emerging sensors exploiting Information and Communication Technologies (ICT). Numerical experiments have been conducted with a real network, and the results have been analyzed from two different perspectives: performance and solution quality. This allows understanding the behavior of the SPSA algorithm and new variants, which altogether contribute to the aim of adding ICT measures in the future. Their sensitivity to the initial values, the effect of bounding the variables and scaling techniques are going to be analyzed. This paper will report on the results of the numerical experiments, their analyses, conclusions and further research.

We establish sharp higher-order Holder regularity estimates up to the boundary for solutions to equations of the form $$\partial _tu-Lu=f(t,x)$$ in $$I\times \Omega $$ where $$I\subset \mathbb {R}$$ , $$\Omega \subset \mathbb {R}^n$$ and f is Holder continuous. The nonlocal operators L that we consider are those arising in stochastic processes with jumps, such as the fractional Laplacian $$(-\Delta )^s$$ , $$s\in (0,1)$$ . Our main result establishes that, if f is $$C^\gamma $$ is space and $$C^{\gamma /2s}$$ in time, and $$\Omega $$ is a $$C^{2,\gamma }$$ domain, then $$u/d^s$$ is $$C^{s+\gamma }$$ up to the boundary in space and u is $$C^{1+\gamma /2s}$$ up the boundary in time, where d is the distance to $$\partial \Omega $$ . This is the first higher order boundary regularity estimate for nonlocal parabolic equations, and is new even for the fractional Laplacian in $$C^\infty $$ domains.

We study the interior regularity of solutions to the Dirichlet problem L u = g in Ω, u = 0 in R n ∖ Ω , for anisotropic operators of fractional type L u ( x ) = ∫ 0 + ∞ d ρ ∫ S n − 1 d a ( ω ) 2 u ( x ) − u ( x + ρ ω ) − u ( x − ρ ω ) ρ 1 + 2 s . Here, a is any measure on S n − 1 (a prototype example for L is given by the sum of one-dimensional fractional Laplacians in fixed, given directions). When a ∈ C ∞ ( S n − 1 ) and g is C ∞ ( Ω ) , solutions are known to be C ∞ inside Ω (but not up to the boundary). However, when a is a general measure, or even when a is L ∞ ( S n − 1 ) , solutions are only known to be C 3 s inside Ω. We prove here that, for general measures a, solutions are C 1 + 3 s − ϵ inside Ω for all ϵ > 0 whenever Ω is convex. When a ∈ L ∞ ( S n − 1 ) , we show that the same holds in all C 1 , 1 domains. In particular, solutions always possess a classical first derivative. The assumptions on the domain are sharp, since if the domain is not convex and the measure a is singular, we construct an explicit counterexample for which u is not C 3 s + ϵ for any ϵ > 0 – even if g and Ω are C ∞ .

Motivated by its relation to models of flame propagation, we study globally Lipschitz solutions of \begin{document}$ \Delta u = f(u) $\end{document} in \begin{document}$ \mathbb{R}^n $\end{document} , where \begin{document}$ f $\end{document} is smooth, non-negative, with support in the interval \begin{document}$ [0,1] $\end{document} . In such setting, any "blow-down" of the solution \begin{document}$ u $\end{document} will converge to a global solution to the classical one-phase free boundary problem of Alt–Caffarelli. In analogy to a famous theorem of Savin for the Allen–Cahn equation, we study here the 1D symmetry of solutions \begin{document}$ u $\end{document} that are energy minimizers. Our main result establishes that, in dimensions \begin{document}$ n , if \begin{document}$ u $\end{document} is axially symmetric and stable then it is 1D.

We study the regularity up to the boundary of solutions to fractional heat equation in bounded $C^{1,1}$ domains. More precisely, we consider solutions to $\partial_t u + (-\Delta)^s u=0 \textrm{ in }\Omega,\ t > 0$, with zero Dirichlet conditions in $\mathbb{R}^n\setminus \Omega$ and with initial data $u_0\in L^2(\Omega)$. Using the results of the second author and Serra for the elliptic problem, we show that for all $t>0$ we have $u(\cdot, t)\in C^s(\mathbb{R}^n)$ and $u(\cdot, t)/\delta^s \in C^{s-\epsilon}(\overline\Omega)$ for any $\epsilon > 0$ and $\delta(x) = \textrm{dist}(x,\partial\Omega)$. Our regularity results apply not only to the fractional Laplacian but also to more general integro-differential operators, namely those corresponding to stable L\'evy processes. As a consequence of our results, we show that solutions to the fractional heat equation satisfy a Pohozaev-type identity for positive times., Comment: This work is part of the bachelor's degree thesis of the first author