Showing papers in "Nonlinear Analysis-theory Methods & Applications in 2022"
TL;DR: For a given p ∈ [ 2, + ∞ ], the authors in this article proved the smooth convergence of the flow for p = 2 in the Euclidean space R n, in the hyperbolic plane H 2, and in the two-dimensional sphere S 2, which implies that such flow in R n or H 2 remains in a bounded region of the space for any time.
Abstract: For a given p ∈ [ 2 , + ∞ ) , we define the p -elastic energy E of a closed curve γ : S 1 → M immersed in a complete Riemannian manifold ( M , g ) as the sum of the length of the curve and the L p -norm of its curvature (with respect to the length measure). We are interested in the convergence of the ( L p , L p ′ ) -gradient flow of these energies to critical points. By means of parabolic estimates, it is usually possible to prove sub-convergence of the flow, that is, convergence to critical points up to reparametrizations and, more importantly, up to isometry of the ambient. Assuming that the flow sub-converges, we are interested in proving the smooth convergence of the flow, that is, the existence of the full limit of the evolving flow. We first give an overview of the general strategy one can apply for proving such a statement. The crucial step is the application of a Łojasiewicz–Simon gradient inequality, of which we present a versatile version. Then we apply such strategy to the flow of E of curves into manifolds, proving the desired improvement of sub-convergence to full smooth convergence of the flow to critical points. As corollaries, we obtain the smooth convergence of the flow for p = 2 in the Euclidean space R n , in the hyperbolic plane H 2 , and in the two-dimensional sphere S 2 . In particular, the result implies that such flow in R n or H 2 remains in a bounded region of the space for any time.
14 citations
TL;DR: In this article , the authors investigate simultaneous recovery inverse problems for semilinear elliptic equations with partial data, based on higher order linearization and monotonicity approaches, and determine the diffusion and absorption coefficients together with the shape of a cavity simultaneously by knowing the corresponding localized Dirichlet-Neumann operator.
Abstract: In this short note, we investigate simultaneous recovery inverse problems for semilinear elliptic equations with partial data. The main technique is based on higher order linearization and monotonicity approaches. With these methods at hand, we can determine the diffusion and absorption coefficients together with the shape of a cavity simultaneously by knowing the corresponding localized Dirichlet–Neumann operator.
14 citations
TL;DR: In this paper, the authors provide a characterization for the weak lower semicontinuity of integral functionals that depend on Riesz fractional gradients instead of ordinary gradients and are considered subject to a complementary-value condition.
Abstract: Based on recent developments in the theory of fractional Sobolev spaces, an interesting new class of nonlocal variational problems has emerged in the literature. These problems, which are the focus of this work, involve integral functionals that depend on Riesz fractional gradients instead of ordinary gradients and are considered subject to a complementary-value condition. With the goal of establishing a comprehensive existence theory, we provide a full characterization for the weak lower semicontinuity of these functionals under suitable growth assumptions on the integrands. In doing so, we surprisingly identify quasiconvexity, which is intrinsic to the standard vectorial calculus of variations, as the natural notion also in the fractional setting. In the absence of quasiconvexity, we determine a representation formula for the corresponding relaxed functionals, obtained via partial quasiconvexification outside the region where complementary values are prescribed. Thus, in contrast to classical results, the relaxation process induces a structural change in the functional, turning the integrand from a homogeneous into an inhomogeneous one. Our proofs rely crucially on an inherent relation between classical and fractional gradients, which we extend to Sobolev spaces, enabling us to transition between the two settings.
10 citations
TL;DR: In this paper , global existence results are provided for a class of cross diffusion equations arising from the modeling of chemotaxis with local sensing, possibly featuring a growth term of logistic-type as well.
Abstract: New estimates and global existence results are provided for a class of systems of cross diffusion equations arising from the modeling of chemotaxis with local sensing, possibly featuring a growth term of logistic-type as well. For sublinear non-increasing motility functions, convergence to the spatially homogeneous steady state is shown, a dedicated Lyapunov functional being constructed for that purpose.
10 citations
TL;DR: In this paper , the authors give an approach to the local Lipschitz continuity of weak solutions to a class of nonlinear elliptic partial differential equations in divergence form of the type ∑i=1n∂∂xiaix,u,Du=bx,u and Duunder p,q−growth assumptions.
Abstract: This article is dedicated to Emmanuele Di Benedetto, great mathematician, colleague, friend. In the spirit to treat a subject that in the last years attracted the interest of several mathematicians, and the attention of Emmanuele too, in this paper we give a first approach to the local Lipschitz continuity of weak solutions to a class of nonlinear elliptic partial differential equations in divergence form of the type ∑i=1n∂∂xiaix,u,Du=bx,u,Duunder p,q−growth assumptions. The novelties with respect to the mathematical literature on this topic are the general growth conditions and the explicit dependence of the differential equation on u, other than on its gradient Du and on the x variable.
9 citations
TL;DR: In this article, the authors considered a double phase problem in R N − div with an unbounded potential V and reaction term f, which does not satisfy the Ambrosetti-Rabinowitz condition.
Abstract: We consider a double phase problem in R N − div ( | ∇ u | p − 2 ∇ u + a ( x ) | ∇ u | q − 2 ∇ u ) + V ( x ) ( | u | p − 2 u + a ( x ) | u | q − 2 u ) = f ( x , u ) with an unbounded potential V and reaction term f , which does not satisfy the Ambrosetti–Rabinowitz condition. A new functional setting was provided for this problem. Using the Fountain and Dual Fountain Theorem with Cerami condition, we obtain some existence of infinitely many solutions. Our result extends some recent work in the literature.
9 citations
TL;DR: In this paper , the forward and inverse problems for the fractional semilinear elliptic equation (−Δ)su+a(x,u)=0 for 0 < s < 1 were investigated.
Abstract: This paper is concerned with the forward and inverse problems for the fractional semilinear elliptic equation (−Δ)su+a(x,u)=0 for 0
8 citations
TL;DR: In this paper , the authors considered a double phase problem with an unbounded potential V and reaction term f, which does not satisfy the Ambrosetti-Rabinowitz condition and obtained some existence of infinitely many solutions.
Abstract: We consider a double phase problem in RN −div(|∇u|p−2∇u+a(x)|∇u|q−2∇u)+V(x)(|u|p−2u+a(x)|u|q−2u)=f(x,u)with an unbounded potential V and reaction term f, which does not satisfy the Ambrosetti–Rabinowitz condition. A new functional setting was provided for this problem. Using the Fountain and Dual Fountain Theorem with Cerami condition, we obtain some existence of infinitely many solutions. Our result extends some recent work in the literature.
8 citations
TL;DR: In this article , the authors consider a class of gradient systems with dissipation potentials of hyperbolic-cosine type and show how such potentials emerge in large deviations of jump processes, multi-scale limits of diffusion processes, and more.
Abstract: We review a class of gradient systems with dissipation potentials of hyperbolic-cosine type. We show how such dissipation potentials emerge in large deviations of jump processes, multi-scale limits of diffusion processes, and more. We show how the exponential nature of the cosh derives from the exponential scaling of large deviations and arises implicitly in cell problems in multi-scale limits. We discuss in-depth the role of "tilting" of gradient systems. Certain classes of gradient systems are "tilt-independent", which means that changing the driving functional does not lead to changes of the dissipation potential. Such tilt-independence separates the driving functional from the dissipation potential, guarantees a clear modelling interpretation, and gives rise to strong notions of gradient-system convergence. We show that although in general many gradient systems are tilt-independent, certain cosh-type systems are not. We also show that this is inevitable, by studying in detail the classical example of the Kramers high-activation-energy limit, in which a diffusion converges to a jump process and the Wasserstein gradient system converges to a cosh-type system. We show and explain how the tilt-independence of the pre-limit system is lost in the limit system. This same lack of independence can be recognized in classical theories of chemical reaction rates in the chemical-engineering literature. We illustrate a similar lack of tilt-independence in a discrete setting. For a class of "two-terminal" fast subnetworks, we give a complete characterization of the dependence on the tilting, which strongly resembles the classical theory of equivalent electrical networks.
8 citations
TL;DR: In this article , the authors studied the regularity and topological properties of volume constrained minimizers of quasi-perimeters in RCD spaces where the reference measure is the Hausdorff measure.
Abstract: In this paper we study regularity and topological properties of volume constrained minimizers of quasi-perimeters in $\sf RCD$ spaces where the reference measure is the Hausdorff measure. A quasi-perimeter is a functional given by the sum of the usual perimeter and of a suitable continuous term. In particular, isoperimetric sets are a particular case of our study. We prove that on an ${\sf RCD}(K,N)$ space $({\rm X},{\sf d},\mathcal{H}^N)$, with $K\in\mathbb R$, $N\geq 2$, and a uniform bound from below on the volume of unit balls, volume constrained minimizers of quasi-perimeters are open bounded sets with $(N-1)$-Ahlfors regular topological boundary coinciding with the essential boundary. The proof is based on a new Deformation Lemma for sets of finite perimeter in ${\sf RCD}(K,N)$ spaces $({\rm X},{\sf d},\mathfrak m)$ and on the study of interior and exterior points of volume constrained minimizers of quasi-perimeters. The theory applies to volume constrained minimizers in smooth Riemannian manifolds, possibly with boundary, providing a general regularity result for such minimizers in the smooth setting.
8 citations
TL;DR: In this article , the existence and behavior of blow-up patterns is split into different regimes by the critical exponent σc and also depends strongly on whether the dimension N≥4 or N∈{2,3}.
Abstract: We study the separate variable blow-up patterns associated to the following second order reaction–diffusion equation: ∂tu=Δum+|x|σum,posed for x∈RN, t≥0, where m>1, dimension N≥2 and σ>0. A new and explicit critical exponent σc=2(m−1)(N−1)3m+1is introduced and a classification of the blow-up profiles is given. The most interesting contribution of the paper is showing that existence and behavior of the blow-up patterns is split into different regimes by the critical exponent σc and also depends strongly on whether the dimension N≥4 or N∈{2,3}. These results extend previous works of the authors in dimension N=1.
TL;DR: In this paper , the authors studied the existence and nonexistence of weak solutions to ∂ku∂tk+(−Δ)mu≥(K∗|u|p)|u|qinRN×R+,∂iu∂ti(x,0)=ui(x)inRN,0≤i≤k−1.
Abstract: We are concerned with the study of existence and nonexistence of weak solutions to ∂ku∂tk+(−Δ)mu≥(K∗|u|p)|u|qinRN×R+,∂iu∂ti(x,0)=ui(x)inRN,0≤i≤k−1, where N,k,m≥1 are positive integers, p,q>0 and ui∈Lloc1(RN) for 0≤i≤k−1. We assume that K is a radial positive and continuous function which decreases in a neighbourhood of infinity. In the above problem, K∗|u|p denotes the standard convolution operation between K(|x|) and |u|p. We obtain necessary conditions on N,m,k,p and q such that the above problem has solutions. Our analysis emphasizes the role played by the sign of ∂k−1u∂tk−1.
TL;DR: In this article , the authors considered a Cahn-Hilliard model with kinetic rate dependent dynamic boundary conditions that was introduced by Knopf et al. (2021) and will thus be called the KLLM model.
Abstract: We consider a Cahn–Hilliard model with kinetic rate dependent dynamic boundary conditions that was introduced by Knopf et al. (2021) and will thus be called the KLLM model. In the aforementioned paper, it was shown that solutions of the KLLM model converge to solutions of the GMS model proposed by Goldstein et al. (2011) as the kinetic rate tends to infinity. We first collect the weak well-posedness results for both models and we establish some further essential properties of the weak solutions. Afterwards, we investigate the long-time behavior of the KLLM model. We first prove the existence of a global attractor as well as convergence to a single stationary point. Then, we show that the global attractor of the GMS model is stable with respect to perturbations of the kinetic rate. Eventually, we construct exponential attractors for both models, and we show that the exponential attractor associated with the GMS model is robust against kinetic rate perturbations.
TL;DR: In this paper , the authors prove the existence of geodesics of probability measures on M which satisfy the entropic semiconvexity inequality defining wTCDpe(K,N) and whose densities with respect to m are additionally uniformly L∞ in time.
Abstract: Let (M,d,m,≪,≤,τ) be a causally closed, K-globally hyperbolic, regular measured Lorentzian geodesic space satisfying the weak timelike curvature-dimension condition wTCDpe(K,N) in the sense of Cavalletti and Mondino. We prove the existence of geodesics of probability measures on M which satisfy the entropic semiconvexity inequality defining wTCDpe(K,N) and whose densities with respect to m are additionally uniformly L∞ in time. This holds apart from any nonbranching assumption. We also discuss similar results under the timelike measure-contraction property.
TL;DR: In this paper , the authors present some basic ideas and techniques in the spectral analysis of lattice Schrödinger operators with disordered potentials, i.e., those given by evaluating a function along an orbit of some ergodic transformation or of several commuting such transformations on higher-dimensional lattices.
Abstract: These lectures present some basic ideas and techniques in the spectral analysis of lattice Schrödinger operators with disordered potentials. In contrast to the classical Anderson tight binding model, the randomness is also allowed to possess only finitely many degrees of freedom. This refers to dynamically defined potentials, i.e., those given by evaluating a function along an orbit of some ergodic transformation (or of several commuting such transformations on higher-dimensional lattices). Classical localization theorems by Fröhlich–Spencer for large disorders are presented, both for random potentials in all dimensions, as well as even quasi-periodic ones on the line. After providing the needed background on subharmonic functions, we then discuss the Bourgain–Goldstein theorem on localization for quasiperiodic Schrödinger cocycles assuming positive Lyapunov exponents.
TL;DR: In this article , it was shown that the Benjamin-Ono equation admits an analytic Birkhoff normal form in an open neighborhood of zero in H0s(T,R) for any s>−1/2 where H 0s(R) denotes the subspace of the Sobolev space of elements with mean 0.
Abstract: In this paper we prove that the Benjamin–Ono equation admits an analytic Birkhoff normal form in an open neighborhood of zero in H0s(T,R) for any s>−1/2 where H0s(T,R) denotes the subspace of the Sobolev space Hs(T,R) of elements with mean 0. As an application we show that for any −1/2
TL;DR: In this article , the authors studied the obstacle problem related to a wide class of nonlinear integro-differential operators, whose model is the fractional sub-Laplacian in the Heisenberg group.
Abstract: We study the obstacle problem related to a wide class of nonlinear integro-differential operators, whose model is the fractional subLaplacian in the Heisenberg group. We prove both the existence and uniqueness of the solution, and that solutions inherit regularity properties of the obstacle such as boundedness, continuity and Hölder continuity up to the boundary. We also prove some independent properties of weak supersolutions to the class of problems we are dealing with. Armed with the aforementioned results, we finally investigate the Perron–Wiener–Brelot generalized solution by proving its existence for very general boundary data.
TL;DR: Optimal regularity estimates for the gradient of solutions to non-uniformly elliptic equations of Orlicz double phase with variable exponents type in divergence form under sharp conditions on such highly nonlinear operators for the Calderón-Zygmund theory were established in this paper .
Abstract: Optimal regularity estimates are established for the gradient of solutions to non-uniformly elliptic equations of Orlicz double phase with variable exponents type in divergence form under sharp conditions on such highly nonlinear operators for the Calderón–Zygmund theory.
TL;DR: In this article, the authors established the boundedness properties of the Neumann extension operator in half-space in the setting of Morrey-Lorentz spaces and derived estimates on the restriction operator in block spaces.
Abstract: We establish the boundedness properties of the Neumann extension operator in half-space in the setting of Morrey–Lorentz spaces. As a by-product we derive estimates on the restriction operator in block spaces. Direct application to solvability of a fourth order nonlinear equation related to higher order boundary conformally invariant problem is considered. By employing a nonvariational approach, we obtain a unique solution under suitable smallness conditions on the boundary data and potentials prescribed in Morrey–Lorentz spaces which allow for singular functions. Moreover, these solutions are shown to be C ∞ in the interior, C l o c 2 , μ ( R + n + 1 ¯ ) , μ ∈ ( 0 , 1 ) in some special case and satisfy interesting qualitative properties including positivity. The results are extended to a larger class of problems involving the polyharmonic operator. In particular, the higher order nonlocal equation ( − Δ ) 2 m − 1 2 v = K ( x ) | v | σ − 1 v + H ( x ) v + g in R n , m ∈ N for 2 ≤ 2 m n + 1 is solvable in a suitable Morrey–Lorentz space and admits positive solutions whenever σ > n / ( n − 2 m + 1 ) . We also prove that no positive solution in the latter framework exists if σ ≤ n / ( n − 2 m + 1 ) .
TL;DR: In this paper , the Cauchy problem for the Hardy-Hénon parabolic equation is studied in the critical and subcritical weighted Lebesgue spaces on the Euclidean space R d .
Abstract: The Cauchy problem for the Hardy–Hénon parabolic equation is studied in the critical and subcritical weighted Lebesgue spaces on the Euclidean space R d . In earlier works, the well-posedness of singular initial data and the existence of non-radial forward self-similar solutions to the problem were shown for the Hardy and Fujita cases ( γ ≤ 0 ). The weighted spaces are used to treat the potential | x | γ as an increase or decrease in the weight, which enables us to prove the well-posedness of the problem for all γ , with − min { 2 , d } < γ , including the Hénon case ( γ > 0 ). As a by-product of global existence, self-similar solutions to the problem are established for all γ without restrictions. Furthermore, the non-existence of a local solution for supercritical data is also shown. Therefore, our critical exponent, s c is optimal with regard to solvability.
TL;DR: In this article, the authors considered a Cahn-Hilliard model with kinetic rate dependent dynamic boundary conditions that was introduced by Knopf et al. (2021) and will thus be called the KLLM model.
Abstract: We consider a Cahn–Hilliard model with kinetic rate dependent dynamic boundary conditions that was introduced by Knopf et al. (2021) and will thus be called the KLLM model. In the aforementioned paper, it was shown that solutions of the KLLM model converge to solutions of the GMS model proposed by Goldstein et al. (2011) as the kinetic rate tends to infinity. We first collect the weak well-posedness results for both models and we establish some further essential properties of the weak solutions. Afterwards, we investigate the long-time behavior of the KLLM model. We first prove the existence of a global attractor as well as convergence to a single stationary point. Then, we show that the global attractor of the GMS model is stable with respect to perturbations of the kinetic rate. Eventually, we construct exponential attractors for both models, and we show that the exponential attractor associated with the GMS model is robust against kinetic rate perturbations.
TL;DR: In this paper, a comparison principle for higher order nonlinear hypoelliptic heat operators on graded Lie groups is presented, and blow-up type results and global in t -boundedness of solutions of nonlinear equations for the heat p -sub-Laplacian on stratified Lie groups are obtained.
Abstract: In this paper we present a comparison principle for higher order nonlinear hypoelliptic heat operators on graded Lie groups. Moreover, using the comparison principle we obtain blow-up type results and global in t -boundedness of solutions of nonlinear equations for the heat p -sub-Laplacian on stratified Lie groups.
TL;DR: In this paper, the authors studied the two-dimensional least gradient problem in a convex polygonal set in the plane and showed existence of solutions when the boundary data are attained in the trace sense.
Abstract: We study the two dimensional least gradient problem in a convex polygonal set in the plane. We show existence of solutions when the boundary data are attained in the trace sense. Due to the lack of strict convexity, the classical results are not applicable. We state the admissibility conditions on the continuous boundary datum f that are sufficient for establishing an existence and uniqueness result. The solutions are constructed by a limiting process, which uses the well-known geometry of superlevel sets of least gradient functions.
TL;DR: In this paper, the authors studied the magnetic trajectories in the generalized Heisenberg group H ( n, 1 ) of dimension ( 2 n + 1 ) endowed with its quasi-Sasakian structure and proved that the trajectories are Frenet curves of maximum order 5.
Abstract: We study the magnetic trajectories in the generalized Heisenberg group H ( n , 1 ) of dimension ( 2 n + 1 ) endowed with its quasi-Sasakian structure. We prove that the trajectories are Frenet curves of maximum order 5 and we completely classify them.
TL;DR: In this article , the critical exponent for the existence of global small data solutions for supercritical powers α>α and do not exist for subcritical powers 1<α<α̃ was established.
Abstract: In this paper, we find the critical exponent for the existence of global small data solutions to: utt+(−Δ)σu+(−Δ)θ2ut=f(u,ut),t≥0,x∈Rn,(u,ut)(0,x)=(0,u1(x)),in the case of so-called non-effective damping, θ∈(σ,2σ], where σ≠1 and f=|u|α or f=|ut|α, in low space dimension. By critical exponent we mean that global small data solution exists for supercritical powers α>α̃ and do not exist, in general, for subcritical powers 1<α<α̃. Assuming initial data to be small in L1 or in some other Lp space, p∈(1,2), in addition to the energy space, the critical exponent only depends on the ratio n/(σp). We also prove the global existence of small data solutions in high space dimension for α>ᾱ, but we leave open to determine if a counterpart nonexistence result for α<ᾱ holds or not.
TL;DR: In this paper, the critical exponent for the existence of global small data solutions for supercritical powers α > α and subcritical powers 1 α α has been established for the case of non-effective damping.
Abstract: In this paper, we find the critical exponent for the existence of global small data solutions to: u t t + ( − Δ ) σ u + ( − Δ ) θ 2 u t = f ( u , u t ) , t ≥ 0 , x ∈ R n , ( u , u t ) ( 0 , x ) = ( 0 , u 1 ( x ) ) , in the case of so-called non-effective damping, θ ∈ ( σ , 2 σ ] , where σ ≠ 1 and f = | u | α or f = | u t | α , in low space dimension. By critical exponent we mean that global small data solution exists for supercritical powers α > α and do not exist, in general, for subcritical powers 1 α α . Assuming initial data to be small in L 1 or in some other L p space, p ∈ ( 1 , 2 ) , in addition to the energy space, the critical exponent only depends on the ratio n / ( σ p ) . We also prove the global existence of small data solutions in high space dimension for α > α , but we leave open to determine if a counterpart nonexistence result for α α holds or not.
TL;DR: In this article , the uniqueness and regularity of solutions in the so-called energy class with sufficiently small energy was studied. But the authors did not address the existence of solutions and certain bubbling phenomena.
Abstract: In this paper, we study the fractional harmonic gradient flow on $S^1$ taking values in $S^{n-1} \subset \mathbb{R}^n$ for every $n \geq 2$, in particular addressing uniqueness and regularity of solutions in the so-called energy class with sufficiently small energy, adding to the existing body of knowledge which includes existence of solutions and certain bubbling phenomena.
TL;DR: In this paper , the existence of multiple solutions for a family of nonlinear elliptic systems in divergence form coupled with a pointwise gradient constraint was proved. But the authors did not consider the nonlinear elasticity of the system.
Abstract: The aim of this paper is to prove the existence of multiple solutions for a family of nonlinear elliptic systems in divergence form coupled with a pointwise gradient constraint: div{A(|x|,|u|2,|∇u|2)∇u}+B(|x|,|u|2,|∇u|2)u=div{P(x)[cof∇u]}inΩ,det∇u=1inΩ,u=φon∂Ω,where Ω⊂Rn (n≥2) is a bounded domain, u=(u1,…,un) is a vector-map and φ is a prescribed boundary condition. Moreover P is a hydrostatic pressure associated with the constraint det∇u≡1 and A=A(|x|,|u|2,|∇u|2), B=B(|x|,|u|2,|∇u|2) are sufficiently regular scalar-valued functions satisfying suitable growths at infinity. The system arises in diverse areas, e.g., in continuum mechanics and nonlinear elasticity, as well as geometric function theory to name a few and a clear understanding of the form and structure of the solutions set is of great significance. The geometric type of solutions constructed here draw upon intimate links with the Lie group SO(n), its Lie exponential and the multi-dimensional curl operator acting on certain vector fields. Most notably, a discriminant type quantity Δ=Δ(A,B) prompting from the system, will be shown to have a decisive role on the structure and multiplicity of these solutions.
TL;DR: The existence and uniqueness theorem for solutions of Hessian equations with prescribed asymptotic behavior at infinity was established in this paper , and a Liouville type result for k-convex solutions was shown in this paper.
Abstract: In this paper, we establish the existence and uniqueness theorem for entire solutions of Hessian equations with prescribed asymptotic behavior at infinity. This extends the previous results on Monge–Ampère equations. Our approach also makes the prescribed asymptotic order optimal within the range preset in exterior Dirichlet problems. In addition, we show a Liouville type result for k-convex solutions. This partly removes the (k+1)- or n-convexity restriction imposed in existing work.