Journal: Nonlinear Analysis

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Elsevier

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0362-546X
1873-5215

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Publications1 - 10 of 22
  • Morse index stability for Sacks–Uhlenbeck approximations for harmonic maps into a sphere
    Item type: Journal Article
    Da Lio, Francesca; Rivière, Tristan; Schlagenhauf, Dominik (2026)
    Nonlinear Analysis
    In this paper we consider sequences of p-harmonic maps, p>2, from a closed Riemann surface Σ into the n-dimensional sphere Sn with uniform bounded energy. These are critical points of the energy Ep(u)≔∫Σ1+|∇u|2p/2dvolΣ. Our two main results are an improved pointwise estimate of the gradient in the neck regions around blow up points and the proof that the necks are asymptotically not contributing to the negativity of the second variation of the energy Ep. This allows us, in the spirit of the paper of the first and second authors in collaboration with Gianocca et al. (2022) , to show the upper semicontinuity of the Morse index plus nullity for sequences of p-harmonic maps into a sphere.
  • Spatial Sobolev regularity for stochastic Burgers equations with additive trace class noise
    Item type: Journal Article
    Jentzen, Arnulf; Lindner, Felix; Pušnik, Primož (2021)
    Nonlinear Analysis
    In this article we investigate the spatial Sobolev regularity of mild solutions to stochastic Burgers equations with additive trace class noise. Our findings are based on a combination of suitable bootstrap-type arguments and a detailed analysis of the nonlinearity in the equation.
  • Well-posedness and inverse problems for semilinear nonlocal wave equations
    Item type: Journal Article
    Lin, Yi-Hsuan; Tyni, Teemu; Zimmermann, Philipp (2024)
    Nonlinear Analysis
    This article is devoted to forward and inverse problems associated with time-independent semilinear nonlocal wave equations. We first establish comprehensive well-posedness results for some semilinear nonlocal wave equations. The main challenge is due to the low regularity of the solutions of linear nonlocal wave equations. We then turn to an inverse problem of recovering the nonlinearity of the equation. More precisely, we show that the exterior Dirichlet-to-Neumann map uniquely determines homogeneous nonlinearities of the form f(x,u) under certain growth conditions. On the other hand, we also prove that initial data can be determined by using passive measurements under certain nonlinearity conditions. The main tools used for the inverse problem are the unique continuation principle of the fractional Laplacian and a Runge approximation property. The results hold for any spatial dimension n∈N.
  • Renormalized powers of Ornstein-Uhlenbeck processes and well-posedness of stochastic Ginzburg-Landau equations
    Item type: Journal Article
    Weinan, E.; Jentzen, Arnulf; Shen, Hao (2016)
    Nonlinear Analysis
  • Variational integrals on Hessian spaces: Partial regularity for critical points
    Item type: Journal Article
    Bhattacharya, Arunima; Skorobogatova, Anna (2025)
    Nonlinear Analysis
    We develop regularity theory for critical points of variational integrals defined on Hessian spaces of functions on open, bounded subdomains of Rn, under compactly supported variations. We show that for smooth convex functionals, a W2,∞ critical point with bounded Hessian is smooth provided that its Hessian has a small bounded mean oscillation (BMO). We deduce that the interior singular set of a critical point has Hausdorff dimension at most n−p0, for some p0∈(2,3). We state some applications of our results to variational problems in Lagrangian geometry. Finally, we use the Hamiltonian stationary equation to demonstrate the importance of our assumption on the a priori regularity of the critical point.
  • Confinement in nonlocal interaction equations
    Item type: Journal Article
    Carrillo, José A.; DiFrancesco, Marco; Figalli, Alessio; et al. (2012)
    Nonlinear Analysis
  • Lipschitz regularity in vectorial linear transmission problems
    Item type: Journal Article
    Figalli, Alessio; Kim, Sunghan; Shahgholian, Henrik (2022)
    Nonlinear Analysis
    We consider vector-valued solutions to a linear transmission problem, and we prove that Lipschitz-regularity on one phase is transmitted to the next phase. More exactly, given a solution u : B$_1$ ⊂ $\mathbb{R}$$^n$ → $\mathbb{R}$$^m$ to the elliptic system div((A+(B−A)$_{χD}$)∇u) = 0 in B$_1$, where A and B are Dini continuous, uniformly elliptic matrices, we prove that if ∇u ∈ L$^∞$(D) then u is Lipschitz in B$_{1/2}$. A similar result is also derived for the parabolic counterpart of this problem.
  • On the Continuity of Center-Outward Distribution and Quantile Functions 
    Item type: Journal Article
    Figalli, Alessio (2018)
    Nonlinear Analysis
  • Stochastic homogenization of the Keller–Segel chemotaxis system
    Item type: Journal Article
    Matzavinos, Anastasios; Ptashnyk, Mariya B. (2016)
    Nonlinear Analysis
  • The Cauchy–Dirichlet problem for the fast diffusion equation on bounded domains
    Item type: Journal Article
    Bonforte, Matteo; Figalli, Alessio (2024)
    Nonlinear Analysis
    The Fast Diffusion Equation (FDE) uₜ = ∆uᵐ, with m ∈ (0,1), is an important model for singular nonlinear (density dependent) diffusive phenomena. Here, we focus on the Cauchy–Dirichlet problem posed on smooth bounded Euclidean domains. In addition to its physical relevance, there are many aspects that make this equation particularly interesting from the pure mathematical perspective. For instance: mass is lost and solutions may extinguish in finite time, merely integrable data can produce unbounded solutions, classical forms of Harnack inequalities (and other regularity estimates) fail to be true, etc. In this paper, we first provide a survey (enriched with an extensive bibliography) focussing on the more recent results about existence, uniqueness, boundedness and positivity (i.e., Harnack inequalities, both local and global), and higher regularity estimates (also up to the boundary and possibly up to the extinction time). We then prove new global (in space and time) Harnack estimates in the subcritical regime. In the last section, we devote a special attention to the asymptotic behaviour, from the first pioneering results to the latest sharp results, and we present some new asymptotic results in the subcritical case.
Publications1 - 10 of 22