Subquadratic Growth Research Articles

Existence of homoclinic solutions for a class of second-order Hamiltonian systems with subquadratic growth

By properly constructing a functional and by using the critical point theory, we establish the existence of homoclinic solutions for a class of subquadratic second-order Hamiltonian systems. Our result generalizes and improves some existing ones. An example is given to show that our theorem applies, while the existing results are not applicable.

Partial regularity for subquadratic parabolic systems with continuous coefficients

We establish the partial regularity of solutions to quasilinear parabolic systems with elliptic part that grows subquadratically. More precisely, it is shown that there is an open subset with full measure, of the solution’s domain, on which the solution is Holder continuous. A key feature in this article is that we only require the coefficients of the system to be continuous with respect to the first two arguments. To prove the result, we use the A-caloric approximation method and an intrinsic scaling. To accommodate the subquadratic growth an adaptation of the A-caloric approximation lemma is also provided.

Partial regularity for subquadratic parabolic systems by $\mathcal{A}$-caloric approximation

We establish a partial regularity result for weak solutions of nonsingular parabolic systems with subquadratic growth of the type \partial_t u - \mathrm{div} a(x,t,u,Du) = B(x,t,u,Du), where the structure function a satisfies ellipticity and growth conditions with growth rate \frac{2n}{n+2} < p < 2 . We prove Hölder continuity of the spatial gradient of solutions away from a negligible set. The proof is based on a variant of a harmonic type approximation lemma adapted to parabolic systems with subquadratic growth.

Optimal partial regularity for sub-elliptic systems with sub-quadratic growth in Carnot groups

This paper is devoted to partial regularity for weak solutions to nonlinear sub-elliptic systems for the case 1 < m < 2 under natural growth conditions in Carnot groups. The method of A -harmonic approximation introduced by Simon and developed by Duzaar, Grotowski and Kronz is adapted to our context, and then partial regularity with the optimal local Hölder exponent for horizontal gradients of weak solutions to the systems is established.

Interior partial regularity for nonlinear elliptic systems with Dini continuous coefficients for the case: [formula omitted

In this article, we consider the interior regularity for weak solutions of nonlinear elliptic systems with divergence type and Dini continuous coefficients under subquadratic growth conditions. We adopt another A -harmonic approximation technique and prove the interior partial regularity of weak solutions. The proof yields directly the optimal Hölder exponent for the derivative of the weak solutions on the regular set.

Optimal partial regularity for nonlinear sub-elliptic systems

In this paper, we consider partial regularity for weak solutions of second-order nonlinear sub-elliptic systems in Carnot groups. By the method of A -harmonic approximation, we establish optimal interior partial regularity of weak solutions to systems under natural growth conditions with subquadratic growth in Carnot groups.

Results on a weighted Poincaré inequality of complete manifolds

We study manifolds satisfying a weighted Poincaré inequality, which was first introduced by Li and Wang. We generalized their result by relaxing the Ricci curvature bound condition only being satisfied outside a compact set and established a finitely many ends result. We also proved a vanishing result for an $L^2$ harmonic 1-form provided that the weight function $\rho$ is of sub-quadratic growth of the distance function, which generalized the Li-Wang result on manifolds with a positive spectrum.

A note on splitting-type variational problems with subquadratic growth

We consider variational problems of splitting-type, i.e., we want to minimize $$ \int\limits_{\Omega}[f(\widetilde{\nabla} w)+g(\partial_{n}w)]\,dx, $$ where $${\widetilde{\nabla}=(\partial_{1},\ldots,\partial_{n-1})}$$ . Here f and g are two C 2-functions which satisfy power growth conditions with exponents 1 < p ≤ q < ∞. In the case p ≥ 2 there is a regularity theory for locally bounded minimizers $${u:\mathbb{R}^n\supset\Omega\rightarrow\mathbb{R}^N}$$ without further restrictions on p and q if n = 2 or N = 1. In the subquadratic case the results are much weaker: we get C 1,α -regularity if we require q ≤ 2p + 2 for n = 2 or q < p + 2 for N = 1. In this paper, we show C 1,α -regularity under the bounds $${q<\frac{2p+4}{2-p}}$$ resp. q < ∞.

Non-linear Calderón–Zygmund theory for parabolic systems with subquadratic growth

For vector-valued solutions of parabolic systems $$\partial_tu-{\rm div}\, a(x,t,Du)={\rm div}\left(|F|^{p-2}F\right)$$ with polynomial growth of rate \({p\in\Big(\frac{2n}{n+2},2\Big)}\), we prove Calderon–Zygmund type estimates for the spatial gradient. In order to deal with the anisotropic scaling behaviour of the above system, we employ the concept of intrinsic geometry by DiBenedetto. Following ideas of Mingione, we avoid tools from harmonic analysis such as singular integrals and maximal functions. Our methods apply to systems that are merely continuous with respect to the space variable as well as to certain systems with a VMO-type regularity. With respect to the time variable, we do not impose any regularity except measurability.

Optimal Interior Partial Regularity for Nonlinear Elliptic Systems for the Case under Natural Growth Condition

We consider the interior regularity for weak solutions of second-order nonlinear elliptic systems with subquadratic growth under natural growth condition. We obtain a general criterion for a weak solution to be regular in the neighborhood of a given point. In particularly the regularity we obtained is optimal.

Global Regularity of Solutions to Systems of Reaction–Diffusion with Sub-Quadratic Growth in Any Dimension

This paper is devoted to the study of the regularity of solutions to some systems of reaction–diffusion equations, with reaction terms having a subquadratic growth. We show the global boundedness and regularity of solutions, without smallness assumptions, in any dimension N. The proof is based on blow-up techniques. The natural entropy of the system plays a crucial role in the analysis. It allows us to use of De Giorgi type methods introduced for elliptic regularity with rough coefficients. In spite these systems are entropy supercritical, it is possible to control the hypothetical blow-ups, in the critical scaling, via a very weak norm. Analogies with the Navier–Stokes equation are briefly discussed in the introduction.

Infinitely many homoclinic solutions for second order Hamiltonian systems

In this paper we study the existence of infinitely many homoclinic solutions for second order Hamiltonian systems u ̈ − L ( t ) u + W u ( t , u ) = 0 , ∀ t ∈ R , where L ( t ) is unnecessarily positive definite for all t ∈ R , and W ( t , u ) is of subquadratic growth as | u | → ∞ .

Partial Hölder continuity for solutions of subquadratic elliptic systems in low dimensions

We consider weak solutions of second order nonlinear elliptic systems in divergence form under standard subquadratic growth conditions with boundary data of class C 1 . In dimensions n ∈ { 2 , 3 } we prove that u is locally Hölder continuous for every exponent λ ∈ ( 0 , 1 − n − 2 p ) outside a singular set of Hausdorff dimension less than n − p . This result holds up to the boundary both for non-degenerate and degenerate systems. In the proof we apply the direct method and classical Morrey-type estimates introduced by Campanato.

Vanishing theorems for ach Kähler manifolds and harmonic maps

We discuss a class of complete Kähler manifolds which are asymptotically complex hyperbolic near infinity. The main result is vanishing theorems for the second L 2 cohomology of such manifolds when it has positive spectrum. We also generalize the result to the weighted Poincaré inequality case and establish a vanishing theorem provided that the weighted function ρ is of sub-quadratic growth of the distance function. We also obtain a vanishing theorem of harmonic maps on manifolds which satisfies the weighted Poincaré inequality.

Partial regularity for weak solutions of nonlinear elliptic systems: the subquadratic case

We consider weak solutions of second order nonlinear elliptic systems of divergence type under subquadratic growth conditions. Via the method of $${\mathcal{A}}$$ -harmonic approximation we give a characterization of regular points up to the boundary which extends known results from the quadratic and superquadratic case. The proof yields directly the optimal higher regularity on the regular set.

Estimates of deviations from minimizers for variational problems with power growth functionals

We derive directly computable estimates of differences between approximate solutions and minimizers of the variational problem $$J_\alpha \left[ w \right]: = \int\limits_\Omega {\left[ {\frac{1}{\alpha }\left| {\nabla w} \right|^\alpha - fw} \right]dx} \to \min .$$ . If the functional has a superquadratic growth, then the estimate is given in terms of the natural energy norm. For problems with subquadratic growth, it is more convenient to derive such estimates in terms of the dual variational problem. The estimates are obtained for Dirichlet, Neumann, and mixed boundary conditions. Bibliography: 22 titles.

Quasilinear elliptic equations with subquadratic growth

In this paper we consider nonlinear boundary value problems whose simplest model is the following: (0.1) { − Δ u + ν | u | p − 1 u = γ | ∇ u | 2 θ + f ( x ) in Ω , u = 0 on ∂ Ω , where f ( x ) is a summable function in Ω (bounded open set in R N , N > 2 ), p > θ ( 1 − θ ) , 0 < θ < 1 and γ ∈ R .

Regularity of almost minimizers of quasi-convex variational integrals with subquadratic growth

We prove a small excess regularity theorem for almost minimizers of a quasi-convex variational integral of subquadratic growth. The proof is direct, and it yields an optimal modulus of continuity for the derivative of the almost minimizer. The result is new for general almost minimizers, and in the case of absolute minimizers it considerably simplifies the existing proof.

Partial regularity of local minimizers of quasiconvex integrals with sub-quadratic growth

We prove partial regularity for local minimizers of quasiconvex integrals of the form I(v) = ∫ΩF(Dv(x))dx, where the integrand f(ξ) has sub-quadratic growth, i.e. |F(ξ)| ≤ L(1 + |ξ|p), with 1 &lt; p &lt; 2. A function u ∈ W1,p(Ω;RN) is a W1,q(Ω;RN) local minimizer of I(v) if there exists a δ &gt; 0 such that I(u) ≤ I(v) whenever v and ‖Dv − Du‖q ≤ δ.

A quadratic Bolza-type problem in a Riemannian manifold

A classical nonlinear equation D s x ̇ + ∇ xV(x,s)=0 on a complete Riemannian manifold M is considered. The existence of solutions connecting any two points x 0,x 1∈ M is studied, i.e., for T>0 the critical points of the functional J T(x)= 1 2 ∫ 0 T 〈 x ̇ , x ̇ 〉 ds− ∫ 0 T V(x,s)ds with x(0)= x 0, x( T)= x 1. When the potential V has a subquadratic growth with respect to x, J T admits a minimum critical point for any T>0 (infinitely many critical points if the topology of M is not trivial). When V has an at most quadratic growth, i.e., V(x,s)⩽λd 2(x, x ̄ )+k , this property does not hold, but an optimal arrival time T( λ)>0 exists such that, if 0< T< T( λ), any pair of points in M can be joined by a critical point of the corresponding functional. For the existence and multiplicity results, variational methods and Ljusternik–Schnirelman theory are used. The optimal value T(λ)=π/ 2λ is fulfilled by the harmonic oscillator. These ideas work for other related problems.