Subquadratic Growth Research Articles

Markovian Quadratic BSDEs with an Unbounded Sub-quadratic Growth

Markovian Quadratic BSDEs with an Unbounded Sub-quadratic Growth

Line-tension limits for line singularities and application to the mixed-growth case

We study variational models for dislocations in three dimensions in the line-tension scaling. We present a unified approach which allows to treat energies with subquadratic growth at infinity and other regularizations of the singularity near the dislocation lines. We show that the asymptotics via Gamma convergence is independent of the specific choice of the energy and of the regularization procedure.

Higher differentiability and existence for a class of problems under p, q subquadratic growth

Higher differentiability and existence for a class of problems under p, q subquadratic growth

Higher differentiability results for solutions to a class of non-homogeneous elliptic problems under sub-quadratic growth conditions

We prove a sharp higher differentiability result for local minimizers of functionals of the form [Formula: see text] with non-autonomous integrand [Formula: see text] which is convex with respect to the gradient variable, under [Formula: see text]-growth conditions, with [Formula: see text]. The main novelty here is that the results are obtained assuming that the partial map [Formula: see text] has weak derivatives in some Lebesgue space [Formula: see text] and the datum [Formula: see text] is assumed to belong to a suitable Lebesgue space [Formula: see text]. We also prove that it is possible to weaken the assumption on the datum [Formula: see text] and on the map [Formula: see text], if the minimizers are assumed to be a priori bounded.

On the optimal Lq-regularity for viscous Hamilton–Jacobi equations with subquadratic growth in the gradient

This paper studies a maximal [Formula: see text]-regularity property for nonlinear elliptic equations of second order with a zeroth order term and gradient nonlinearities having superlinear and subquadratic growth, complemented with Dirichlet boundary conditions. The approach is based on the combination of linear elliptic regularity theory and interpolation inequalities, so that the analysis of the maximal regularity estimates boils down to determine lower order integral bounds. The latter are achieved via a [Formula: see text] duality method, which exploits the regularity properties of solutions to stationary Fokker–Planck equations. For the latter problems, we discuss both global and local estimates. Our main novelties for the regularity properties of this class of nonlinear elliptic boundary-value problems are the analysis of the end-point summability threshold [Formula: see text], [Formula: see text] being the dimension of the ambient space and [Formula: see text] the growth of the first-order term in the gradient variable, along with the treatment of the full integrability range [Formula: see text].

Triangulations of uniform subquadratic growth are quasi-trees

Triangulations of uniform subquadratic growth are quasi-trees

Regularity results for solutions to a class of non-autonomous obstacle problems with sub-quadratic growth conditions

We establish some higher differentiability results for solution to non-autonomous obstacle problems of the form min∫Ωfx,Dv(x)dx:v∈Kψ(Ω),where the function f satisfies p−growth conditions with respect to the gradient variable, for 1<p<2, and Kψ(Ω) is the class of admissible functions.Here we show that, if the obstacle ψ is bounded, then a Sobolev regularity assumption on the gradient of the obstacle ψ transfers to the gradient of the solution, provided the partial map x↦Dξf(x,ξ) belongs to a Sobolev space, W1,p+2.The novelty here is that we deal with subquadratic growth conditions with respect to the gradient variable, i.e. f(x,ξ)≈a(x)|ξ|p with 1<p<2, and where the map a belongs to a Sobolev space.

Beyond the Quadratic Approximation: The Multiscale Structure of Neural Network Loss Landscapes

A quadratic approximation of neural network loss landscapes has been extensively used to study the optimization process of these networks. Though, it usually holds in a very small neighborhood of the minimum, it cannot explain many phenomena observed during the optimization process. In this work, we study the structure of neural network loss functions and its implication on optimization in a region beyond the reach of a good quadratic approximation. Numerically, we observe that neural network loss functions possesses a multiscale structure, manifested in two ways: (1) in a neighborhood of minima, the loss mixes a continuum of scales and grows subquadratically, and (2) in a larger region, the loss shows several separate scales clearly. Using the subquadratic growth, we are able to explain the Edge of Stability phenomenon [5] observed for the gradient descent (GD) method. Using the separate scales, we explain the working mechanism of learning rate decay by simple examples. Finally, we study the origin of the multiscale structure and propose that the non-convexity of the models and the non-uniformity of training data is one of the causes. By constructing a two-layer neural network problem we show that training data with different magnitudes give rise to different scales of the loss function, producing subquadratic growth and multiple separate scales.

Besov regularity for a class of singular or degenerate elliptic equations

Motivated by applications to congested traffic problems, we establish higher integrability results for the gradient of local weak solutions to the strongly degenerate or singular elliptic PDE−div((|∇u|−1)+q−1∇u|∇u|)=f,inΩ, where Ω is a bounded domain in Rn for n≥2, 1<q<∞ and (⋅)+ stands for the positive part. We assume that the datum f belongs to a suitable Sobolev or Besov space. The main novelty here is that we deal with the case of subquadratic growth, i.e. 1<q<2, which has so far been neglected. In the latter case, we also prove the higher fractional differentiability of the solution to a variational problem, which is characterized by the above equation. For the sake of completeness, we finally give a Besov regularity result also in the case q≥2.

Propagation of Wave Packets for Systems Presenting Codimension One Crossings

We analyze the propagation of wave packets through general Hamiltonian systems presenting codimension one eigenvalue crossings. The class of time-dependent Hamiltonians we consider is of general pseudodifferential form with subquadratic growth. It comprises Schrödinger operators with matrix-valued potential, as they occur in quantum molecular dynamics, but also covers matrix-valued models of solid state physics describing the motion of electrons in a crystal. We calculate precisely the non-adiabatic effects of the crossing in terms of a transition operator, whose action on coherent states can be spelled out explicitly.

Higher differentiability results for solutions to a class of non-autonomous obstacle problems with sub-quadratic growth conditions

Abstract We establish some higher differentiability results of integer and fractional order for solutions to non-autonomous obstacle problems of the form min ⁡ { ∫ Ω f ⁢ ( x , D ⁢ v ⁢ ( x ) ) : v ∈ 𝒦 ψ ⁢ ( Ω ) } , \min\biggl{\{}\int_{\Omega}f(x,Dv(x)):v\in\mathcal{K}_{\psi}(\Omega)\biggr{\}}, where the function f satisfies p-growth conditions with respect to the gradient variable, for 1 &lt; p &lt; 2 {1&lt;p&lt;2} , and 𝒦 ψ ⁢ ( Ω ) {\mathcal{K}_{\psi}(\Omega)} is the class of admissible functions v ∈ u 0 + W 0 1 , p ⁢ ( Ω ) {v\in u_{0}+W^{1,p}_{0}(\Omega)} such that v ≥ ψ {v\geq\psi} a.e. in Ω, where u 0 ∈ W 1 , p ⁢ ( Ω ) {u_{0}\in W^{1,p}(\Omega)} is a fixed boundary datum. Here we show that a Sobolev or Besov–Lipschitz regularity assumption on the gradient of the obstacle ψ transfers to the gradient of the solution, provided the partial map x ↦ D ξ ⁢ f ⁢ ( x , ξ ) {x\mapsto D_{\xi}f(x,\xi)} belongs to a suitable Sobolev or Besov space. The novelty here is that we deal with sub-quadratic growth conditions with respect to the gradient variable, i.e. f ⁢ ( x , ξ ) ≈ a ⁢ ( x ) ⁢ | ξ | p {f(x,\xi)\approx a(x)\lvert\xi\rvert^{p}} with 1 &lt; p &lt; 2 {1&lt;p&lt;2} , and where the map a belongs to a Sobolev or Besov–Lipschitz space.

Well-posedness of scalar BSDEs with sub-quadratic generators and related PDEs

We first establish the existence of an unbounded solution to a backward stochastic differential equation (BSDE) with generator g allowing a general growth in the state variable y and a sub-quadratic growth in the state variable z, when the terminal condition satisfies a sub-exponential moment integrability condition, which is weaker than the usual exp(μL)-integrability and stronger than Lp(p>1)-integrability. Then, we prove the uniqueness and comparison theorem for the unbounded solutions of the preceding BSDEs under some additional assumptions and establish a general stability result for the unbounded solutions. Finally, we derive the nonlinear Feynman–Kac formula in this context.

Attractiveness of Constant States in Logistic-Type Keller–Segel Systems Involving Subquadratic Growth Restrictions

Abstract The chemotaxis-growth system ($\star$) { u t = D ⁢ Δ ⁢ u - χ ⁢ ∇ ⋅ ( u ⁢ ∇ ⁡ v ) + ρ ⁢ u - μ ⁢ u α , v t = d ⁢ Δ ⁢ v - κ ⁢ v + λ ⁢ u {}\left\{\begin{aligned} \displaystyle{}u_{t}&amp;\displaystyle=D\Delta u-\chi% \nabla\cdot(u\nabla v)+\rho u-\mu u^{\alpha},\\ \displaystyle v_{t}&amp;\displaystyle=d\Delta v-\kappa v+\lambda u\end{aligned}\right. is considered under homogeneous Neumann boundary conditions in smoothly bounded domains Ω ⊂ ℝ n {\Omega\subset\mathbb{R}^{n}} , n ≥ 1 {n\geq 1} . For any choice of α &gt; 1 {\alpha&gt;1} , the literature provides a comprehensive result on global existence for widely arbitrary initial data within a suitably generalized solution concept, but the regularity properties of such solutions may be rather poor, as indicated by precedent results on the occurrence of finite-time blow-up in corresponding parabolic-elliptic simplifications. Based on the analysis of a certain eventual Lyapunov-type feature of ($\star$), the present work shows that, whenever α ≥ 2 - 2 n {\alpha\geq 2-\frac{2}{n}} , under an appropriate smallness assumption on χ, any such solution at least asymptotically exhibits relaxation by approaching the nontrivial spatially homogeneous steady state ( ( ρ μ ) 1 α - 1 , λ κ ⁢ ( ρ μ ) 1 α - 1 ) {\bigl{(}\bigl{(}\frac{\rho}{\mu}\bigr{)}^{\frac{1}{\alpha-1}},\frac{\lambda}{% \kappa}\bigl{(}\frac{\rho}{\mu}\bigr{)}^{\frac{1}{\alpha-1}}\bigr{)}} in the large time limit.

Periodic Solutions for N-Body-Type Problems

The authors consider non-autonomous N-body-type problems with strong force type potentials at the origin and sub-quadratic growth at infinity. Using Ljusternik-Schnirelmann theory, the authors prove the existence of unbounded sequences of critical values for the Lagrangian action corresponding to non-collision periodic solutions.

Regularity for sub-elliptic systems with VMO-coefficients in the Heisenberg group: the sub-quadratic structure case

AbstractWe consider nonlinear sub-elliptic systems with VMO-coefficients for the case 1 &lt;p&lt; 2 under controllable growth conditions, as well as natural growth conditions, respectively, in the Heisenberg group. On the basis of a generalization of the technique of 𝓐-harmonic approximation introduced by Duzaar-Grotowski-Kronz, and an appropriate Sobolev-Poincaré type inequality established in the Heisenberg group, we prove partial Hölder continuity results for vector-valued solutions of discontinuous sub-elliptic problems. The primary model covered by our analysis is the non-degenerate sub-ellipticp-Laplacian system with VMO-coefficients, involving sub-quadratic growth terms.

Partial Regularity for Stationary Navier-Stokes Systems by the Method of $$\mathcal{A}$$-Harmonic Approximation

In this article, we prove a regularity result for weak solutions away from singular set of stationary Navier-Stokes systems with subquadratic growth under controllable growth condition. The proof is based on the $$\mathcal{A}$$ -harmonic approximation technique. In this article, we extend the result of Shuhong Chen and Zhong Tan [7] and Giaquinta and Modica [18] to the stationary Navier-Stokes system with subquadratic growth.

Notes on Nontrivial Multiple Periodic Solutions for Second-Order Discrete Hamiltonian System

We obtain new existence of multiple nontrivial M-periodic solutions for a class of second-order discrete Hamiltonian system with the force F(t, x) being neither supquadratic nor subquadratic growth in x. Moreover, we exhibit four instructive examples to illustrate our results.

Regularity for minimizers of a class of non-autonomous functionals with sub-quadratic growth

Abstract We consider functionals of the form ℱ ⁢ ( v , Ω ) = ∫ Ω f ⁢ ( x , D ⁢ v ⁢ ( x ) ) ⁢ 𝑑 x , \mathcal{F}(v,\Omega)=\int_{\Omega}f(x,Dv(x))\,dx, with convex integrand with respect to the gradient variable, assuming that the function that measures the oscillation of the integrand with respect to the x variable belongs to a suitable Sobolev space W 1 , q {W^{1,q}} . We prove a higher differentiability result for the minimizers. We also infer a Lipschitz regularity result of minimizers if q &gt; n {q&gt;n} , and a result of higher integrability for the gradient if q = n {q=n} . The novelty here is that we deal with integrands satisfying subquadratic growth conditions with respect to gradient variable.

Partial regularity for discontinuous sub-elliptic systems with subquadratic growth in the Heisenberg group

We consider nonlinear sub-elliptic systems with VMO-coefficients in the Heisenberg group, and prove partial Hölder continuity for discontinuous sub-elliptic problems with subquadratic growth by a generalization of the technique of A-harmonic approximation. The model case is the sub-elliptic p-Laplace system with VMO-coefficients in the form −∑i=12nXiAiα(ξ)1+|Xu|2p−22Xiuα=0,where Aiα(ξ)∈VMO, and 1<p<2.

A semilinear Schrödinger equation with zero on the boundary of the spectrum and exponential growth in ℝ2

The main purpose of this paper is to establish the existence of solutions for the semilinear Schrödinger equation [Formula: see text] where the potential [Formula: see text] is periodic, [Formula: see text] lies on the boundary of a spectral gap of the Schrödinger operator [Formula: see text] and the nonlinearity [Formula: see text] is periodic and has subquadratic exponential growth. The proofs rely on a linking-type argument and a Trudinger–Moser type inequality proved in this paper.