Article
Mathematics
Yinbin Deng, Shuangjie Peng, Xian Yang
Summary: In this paper, the authors study the Choquard equations with fractional Laplacian and establish the uniqueness and non-degeneracy of positive ground states as α approaches 0 and α approaches N, respectively. The paper also provides some uniform regularity and decay estimates for the solutions to the fractional Choquard equation, which are of independent interest and significance.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Heilong Mi, Xiaoqing Deng, Wen Zhang
Summary: This paper investigates a class of fractional p-Laplacian equations with asymptotically periodic conditions, and two new existence results of ground state solutions are obtained under certain suitable conditions using variational methods.
APPLIED MATHEMATICS LETTERS
(2021)
Article
Mathematics, Applied
Abdelrazek Dieb, Isabella Ianni, Alberto Saldana
Summary: This study focuses on positive solutions of a fractional Lane-Emden-type problem in a bounded domain with Dirichlet conditions. It is found that uniqueness and nondegeneracy hold for the asymptotically linear problem in general domains. Furthermore, it is proven that all the known uniqueness and nondegeneracy results in the local case extend to the nonlocal regime when the fractional parameter s is sufficiently close to 1.
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS
(2023)
Article
Mathematics, Applied
Zhenyu Guo, Yan Deng
Summary: The aim of this paper is to study a fractional system with critical non-linearities, and the existence of ground state solutions for the problem is obtained in two cases using the Mountain Pass Theorem.
ANNALS OF FUNCTIONAL ANALYSIS
(2022)
Article
Mathematics, Applied
Yongshuai Gao, Yong Luo
Summary: This paper considers the ground states of a mass subcritical rotational nonlinear Schrödinger equation and shows that they can be described equivalently by minimizers of a constrained variational problem. It proves the existence of minimizers for any rho in the range (0, oo) when 0 < Q < Q* and no minimizers exist for any rho when Q > Q*. The paper also analyzes the mass concentration behavior of minimizers for 0 < Q < Q* as rho approaches infinity. Finally, it proves the existence of a unique minimizer when Q is fixed in the range (0, Q*) and rho is large enough.
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS
(2023)
Article
Mathematics, Applied
Li Ma, Fangyuan Dong
Summary: This paper establishes a new fractional interpolation inequality and compact imbedding result for radially symmetric measurable functions on the whole space. It shows that the best constant in the interpolation inequality can be achieved by a radially symmetric function. As applications, the existence of ground states and standing waves, as well as their orbital stability, are studied for the nonlinear fractional Schrodinger equation on the whole space.
PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS
(2023)
Article
Mathematics, Applied
Xudong Shang
Summary: This work investigates the existence, multiplicity, and concentration behavior of positive solutions involving the fractional p-Laplacian. The number of solutions is related to the topology of the sets where the potential functions V and K attain their global minimum and maximum.
Article
Mathematics, Applied
Luca Vilasi, Youjun Wang
Summary: In this study, we investigate the blow-up behavior of ground states of the fractional Choquard equation as the exponent p(epsilon) approaches the upper critical growth regime. We prove that the ground state blows up in the sense of epsilon approaching zero.
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS
(2022)
Article
Mathematics
Maria Ahrend, Enno Lenzmann
Summary: We prove the uniqueness of solutions for the nonlocal Liouville equation under certain conditions, where the Q-curvature function satisfies certain requirements. Furthermore, we establish a connection between this equation and the ground state solitons for the Calogero-Moser derivative NLS.
JOURNAL OF FUNCTIONAL ANALYSIS
(2022)
Article
Mathematics, Applied
Lars Bugiera, Enno Lenzmann, Jeremy Sok
Summary: This article studies ground state solutions for linear and nonlinear elliptic PDEs in Rn with (pseudo-)differential operators of arbitrary order. In the nonlinear case, a general symmetry result is proven, while a uniqueness result for ground states is proven in the linear case. Instead of traditional methods such as maximum principles, moving planes, or Polya-Szego inequalities, arguments based on the Fourier transform and a recently obtained rigidity result for the Hardy-Littlewood majorant problem are used.
SIAM JOURNAL ON MATHEMATICAL ANALYSIS
(2022)
Article
Physics, Mathematical
Xiaoming An, Xian Yang
Summary: In this paper, we establish a connection between fractional Schrodinger equations with power-law nonlinearity and fractional Schrodinger equations with logarithm-law nonlinearity. We prove that the ground state solutions of power-law fractional equations approach a ground state solution of logarithm-law fractional equations as p approaches 2(+). Furthermore, we provide a novel proof for the existence of a ground state in logarithm-law fractional Schrodinger equations.
JOURNAL OF MATHEMATICAL PHYSICS
(2023)
Article
Mathematics, Applied
Mao-sheng Chang, Chiun-chuan Chen, Yung-ta Li
Summary: In this note, we investigate the backwards in time problem of the heat equation in a fractional setting with a suitable Neumann condition, and establish a result of uniqueness.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B
(2023)
Article
Mathematics, Applied
Giulia Meglioli, Fabio Punzo
Summary: This article investigates the uniqueness of solutions to a class of linear, nonlocal parabolic problems with a drift term in suitable weighted Lebesgue spaces. The problem is nonlocal due to the presence of the fractional Laplacian as diffusion operator. The drift term is driven by a smooth enough, possibly unbounded vector field b, which satisfies a suitable growth condition in the set {x is an element of RN : ⟨b(x), x⟩ > 0}. In general, the uniqueness class includes unbounded solutions, and specifically, uniqueness of bounded solutions is obtained. Furthermore, the hypothesis on the drift term b is shown to be sharp, as the violation of the mentioned growth condition leads to infinitely many bounded solutions. Finally, the uniqueness of a linear, nonlocal elliptic equation with a drift term is also investigated, yielding similar results.
COMMUNICATIONS ON PURE AND APPLIED ANALYSIS
(2023)
Article
Mathematics, Applied
Yong Luo, Shu Zhang
Summary: This paper examines the ground state of a nonlinear Schrödinger equation and investigates its existence and concentration behavior through solving a constrained minimization problem. The study establishes the threshold for the existence of the ground state and proves the uniqueness of the solution when it is close to the threshold.
COMMUNICATIONS ON PURE AND APPLIED ANALYSIS
(2022)
Article
Mathematics, Applied
Jiaojiao Li, Li Ma
Summary: In this paper, extremals of a new Gagliardo-Nirenberg type inequality with symmetry are considered, which are potentially relevant for studying the existence of ground states to a class of nonlinear fractional Schrodinger equations. The paper proves the existence of ground states of the fractional Schrodinger equation under certain restrictions, and demonstrates the existence of first eigen-functions of non-local problems on the whole space.
APPLIED MATHEMATICS LETTERS
(2021)
Article
Mathematics, Applied
Xavier Cabre, Serena Dipierro, Enrico Valdinoci
Summary: In this study, we extend the classical Bernstein technique to integro-differential operators and provide first and one-sided second derivative estimates for solutions to fractional equations. Our method is applicable to Pucci-type extremal equations and obstacle problems for fractional operators. Several results are new even in the linear case. The study also raises some intriguing open questions.
ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
(2022)
Article
Mathematics, Applied
Serena Dipierro, Giorgio Poggesi, Enrico Valdinoci
Summary: In this paper, we prove the radial symmetry for bounded nonnegative solutions for 1 < p < infinity. We extend the symmetry result in the isotropic unweighted setting to the anisotropic setting, which is a generalization of a celebrated result by Gidas-Ni-Nirenberg. The results obtained in this paper are new even for the isotropic and unweighted setting, and we also provide a new approach to the problem by exploiting integral (in)equalities.
CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
(2022)
Article
Mathematics, Applied
Stefano Biagi, Serena Dipierro, Enrico Valdinoci, Eugenio Vecchi
Summary: A systematic study of superpositions of elliptic operators with different orders was conducted, focusing on the sum of the Laplacian and fractional Laplacian. Structural results were provided, including existence, maximum principles for weak and classical solutions, interior Sobolev regularity, and boundary regularity of Lipschitz type.
COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS
(2022)
Article
Mathematics
Claudia Bucur, Serena Dipierro, Luca Lombardini, Jose M. Mazon, Enrico Valdinoci
Summary: We investigate the convergence of the minimizers of the W-s(,p)-energy to those of the W-s(,1)-energy as p tends to 1, both in the pointwise sense and by means of Gamma-convergence. We also study the convergence of the corresponding Euler-Lagrange equations and the equivalence between minimizers and weak solutions. As ancillary results, we examine some regularity issues regarding minimizers of the W-s(,1)-energy.
INTERNATIONAL MATHEMATICS RESEARCH NOTICES
(2023)
Article
Mathematics
Cecilia Cavaterra, Serena Dipierro, Zu Gao, Enrico Valdinoci
Summary: We provide global gradient estimates for solutions to a general type of nonlinear parabolic equations in a possibly Riemannian geometry setting. Our new result is distinguished from the existing ones in the literature by the validity of the estimates in the global domain and the detection of several additional regularity effects resulting from special parabolic data. Additionally, our result encompasses a wide range of nonlinear sources treated through a unified approach and recovers many classical results as special cases.
JOURNAL OF GEOMETRIC ANALYSIS
(2022)
Article
Mathematics, Applied
Serena Dipierro, Fumihiko Onoue, Enrico Valdinoci
Summary: This paper investigates nonlocal minimal surfaces in a cylinder with prescribed datum given by the complement of a slab. It shows that the connectivity of the minimizers is related to the width of the slab, and provides a quantitative bound on the stickiness property exhibited by the minimizers.
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
(2022)
Article
Mathematics, Applied
Serena Dipierro, Aleksandr Dzhugan, Enrico Valdinoci
Summary: We introduce a suitable notion of integral operators (including the fractional Laplacian as a special case) that act on functions with minimal requirements at infinity. To avoid divergent expressions, we propose a cutoff procedure to replace the classical definition. The resulting notion eliminates the polynomials that cause the divergent pattern when the cutoff is removed. We demonstrate stability and compatibility of different-order polynomials under an appropriate notion of convergence, and discuss the solvability of the Dirichlet problem. Furthermore, we present a viscosity counterpart under the additional assumption that the interaction kernel has a sign, consistent with the maximum principle structure.
FRACTIONAL CALCULUS AND APPLIED ANALYSIS
(2022)
Article
Mathematics
Giulio Ciraolo, Serena Dipierro, Giorgio Poggesi, Luigi Pollastro, Enrico Valdinoci
Summary: We study symmetry and quantitative approximate symmetry for an overdetermined problem involving the fractional torsion problem in a bounded open set Ω in Rn. We prove that if the fractional torsion function has a C1 level surface which is parallel to the boundary ∂Ω, then Ω is a ball. If the solution is close to a constant on a parallel surface to the boundary, we quantitatively prove that Ω is close to a ball. Our results use techniques peculiar to the nonlocal case, such as quantitative versions of the fractional Hopf boundary point lemma and boundary Harnack estimates for antisymmetric functions. We also provide an application to the study of rural-urban fringes in population settlements.
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
(2023)
Article
Mathematics
Alessandra De Luca, Serena Dipierro, Enrico Valdinoci
Summary: We investigate a nonlocal capillarity problem, in which the interaction kernels may have anisotropy and not be invariant under scaling. We introduce two fractional exponents S-1 and S-2, which indicate the lack of scale invariance and account for different particle interaction features between the container and environment. We establish a nonlocal Young's law for the contact angle and discuss the unique solvability of the equation in terms of the interaction kernels and relative adhesion coefficient.
MATHEMATISCHE ANNALEN
(2023)
Article
Mathematics
Serena Dipierro, Enrico Valdinoci
Summary: In this paper, we present both classical and modern results on phase transitions and minimal surfaces, showing their deep connection. We start by revisiting Lev Landau's theory of phase transitions. Then, we investigate the relationship between short-range phase transitions and classical minimal surfaces, introducing their regularity theory and Ennio De Giorgi's celebrated conjecture. Finally, we explore long-range phase transitions and their connection to the analysis of fractional minimal surfaces.
BULLETIN OF MATHEMATICAL SCIENCES
(2023)
Article
Mathematics, Applied
Serena Dipierro, Ovidiu Savin, Enrico Valdinoci
Summary: Nonlocal minimal surfaces, unlike their classical counterparts, exhibit discontinuities and adhere to the boundary of smooth domains. This paper confirms the conjecture that stickiness is larger near concave portions of the boundary and absent in corners of the square. The study also shows that the nonlocal minimal surfaces are continuous at convex corners and discontinuous at concave corners of the domain boundary.
CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Serena Dipierro, Aram Karakhanyan, Enrico Valdinoci
Summary: This paper classifies non-trivial, non-negative, positively homogeneous solutions of the equation Au = yuY-1 in the plane. The motivation for this problem comes from the analysis of the classical Alt-Phillips free boundary problem, but with negative exponents y. The proof relies on several custom results for ordinary differential equations.
INTERFACES AND FREE BOUNDARIES
(2023)
Article
Mathematics
Serena Dipierro, Jack Thompson, Enrico Valdinoci
Summary: This paper proves the Harnack inequality for antisymmetric s-harmonic functions and fractional equations with zeroth-order terms in a general domain. This result can be used with the moving planes method to obtain quantitative stability results for symmetry and overdetermined problems for semilinear equations driven by the fractional Laplacian. The proof is divided into two parts: an interior Harnack inequality away from the plane of symmetry and a boundary Harnack inequality close to the plane of symmetry. These results are established by first establishing the weak Harnack inequality for super-solutions and local boundedness for sub-solutions in both the interior and boundary cases. Additionally, a new mean value formula for antisymmetric s-harmonic functions is obtained.
JOURNAL OF FUNCTIONAL ANALYSIS
(2023)
Article
Physics, Multidisciplinary
Serena Dipierro, Giovanni Giacomin, Enrico Valdinoci
Summary: This study investigates the optimal foraging strategies for stationary prey in a given region of space. The research shows that for uniformly distributed prey, the best strategy for the forager is to be stationary and uniformly distributed in the same region. In cases where foragers cannot be completely stationary, the study explores the best seeking strategy for Levy foragers based on the Levy exponent. The findings reveal that the optimal strategy depends on the size of the region where the prey is located, with large regions favoring Gaussian random walks and small regions favoring Levy foragers with small fractional exponent.
JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL
(2023)
Article
Mathematics, Interdisciplinary Applications
Stefano Biagi, Serena Dipierro, Enrico Valdinoci, Eugenio Vecchi
Summary: In this study, we investigate the eigenvalue problem for a nonlinear mixed local/nonlocal operator with vanishing conditions. We demonstrate that the second eigenvalue is always strictly larger than the first eigenvalue of a ball, with the volume of the ball being half that of the given bounded open set. We prove the sharpness of this bound by comparing it to a limit case involving two equal balls far apart.
MATHEMATICS IN ENGINEERING
(2023)
