Harnack Type Research Articles

Harnack type inequality and Liouville theorem for subcritical fully nonlinear equations

We consider this equation σk(Au)=up−n+2n−2k,where n≥3 and p∈nn−2,n+2n−2. Here σk denotes the kth elementary symmetric function of the eigenvalues of Au, and Au=−2n−2u−n+2n−2D2u+2n(n−2)2u−2nn−2∇u⊗∇u−2(n−2)2u−2nn−2|∇u|2I,where ∇u denotes the gradient of u and D2u denotes the Hessian of u. This equation has fruitful backgrounds in geometry and physics. We then obtain Schoen’s Harnack type inequality in Euclidean balls, and asymptotic behavior of an entire solution. Based on the asymptotic behavior, we give another proof of the Liouville theorem obtained by A. Li and Y.Y. Li (2005).

Hölder continuity and Harnack estimate for non-homogeneous parabolic equations

In this paper we continue the study on intrinsic Harnack inequality for non-homogeneous parabolic equations in non-divergence form initiated by the first author in Arya (Calc Var Partial Differ Equ 61:30–31, 2022). We establish a forward-in-time intrinsic Harnack inequality, which in particular implies the Hölder continuity of the solutions. We also provide a Harnack type estimate on global scale which quantifies the strong minimum principle. In the time-independent setting, this together with Arya (2022) provides an alternative proof of the generalized Harnack inequality proven by the second author in Julin (Arch Ration Mech Anal 216:673–702, 2015).

Monostable pulsating traveling waves in discrete periodic media with delay

The purpose of this article is to investigate various qualitative properties of pulsating traveling waves for a delayed lattice dynamical system with global interaction in periodic habitat. Under some appropriate assumptions, we first establish a Harnack type of inequality for its wave profile system. Then, we show that all wave profiles remain strictly increasing in the propagation process. Based on the Harnack’s inequality and monotonicity result, the other effort of the article is devoted to the exponential decay of the non-critical pulsating traveling waves towards the unstable steady state. Finally, we obtain the uniqueness of all non-critical pulsating traveling waves.

Positive radial solutions for Dirichlet problems in the ball

We are concerned with existence, localization and multiplicity of positive radial solutions to Dirichlet problems with ϕ-Laplacians in a ball, in both scalar and system cases. Our approach essentially relies on fixed point index computations and a main feature is that it avoids any Harnack type inequality. Applications to some problems involving operators with Uhlenbeck structure are discussed.

A new Schwarz-Pick lemma at the boundary and rigidity of holomorphic maps

In this paper we establish several invariant boundary versions of the (infinitesimal) Schwarz-Pick lemma for conformal pseudometrics on the unit disk and for holomorphic selfmaps of strongly convex domains in CN in the spirit of the boundary Schwarz lemma of Burns–Krantz. Firstly, we focus on the case of the unit disk and prove a general boundary rigidity theorem for conformal pseudometrics with variable curvature. In its simplest cases this result already includes new types of boundary versions of the lemmas of Schwarz–Pick, Ahlfors–Schwarz and Nehari–Schwarz. The proof is based on a new Harnack–type inequality as well as a boundary Hopf lemma for conformal pseudometrics which extend earlier interior rigidity results of Golusin, Heins, Beardon, Minda and others. Secondly, we prove similar rigidity theorems for sequences of conformal pseudometrics, which even in the interior case appear to be new. For instance, a first sequential version of the strong form of Ahlfors' lemma is obtained. As an auxiliary tool we establish a Hurwitz–type result about preservation of zeros of sequences of conformal pseudometrics. Thirdly, we apply the one–dimensional sequential boundary rigidity results together with a variety of techniques from several complex variables to prove a boundary version of the Schwarz–Pick lemma for holomorphic maps of strongly convex domains in CN for N>1.

Harnack Type Inequalities for SDEs Driven by Fractional Brownian Motion with Markovian Switching

Harnack Type Inequalities for SDEs Driven by Fractional Brownian Motion with Markovian Switching

Local Hessian estimates of solutions to nonlinear parabolic equations along Ricci flow

In the paper, we first obtain some Hessian estimates for any positive solution to the nonlinear parabolic equations$ \begin{equation*} \partial_t u(x, t) = \Delta_{g(t)} u(x, t)+\lambda u^{\alpha}(x, t) \end{equation*} $on a Riemannian manifold with a fixed metric and along the Ricci flow by constructing a new auxiliary function. Moreover, we only need the curvature tensor is bounded. These estimates imply some local, time reversed Harnack type inequalities. And our conclusion generalizes the Han and Zhang's result, but also our proof process is different and has been slightly simplified. As its applications, we derive the Hessian estimate of harmonic function and eigenfunction.

Componentwise localization of solutions to systems of operator inclusions via Harnack type inequalities

We establish compression-expansion type fixed point theorems for systems of operator inclusions with decomposable multivalued maps. The approach is vectorial allowing to localize individually the components of solutions and to obtain multiple solutions with multiplicity not necessarily concerned with all components of the solution. A general scheme of applicability of the theory is elaborated based on Harnack type inequalities and illustrated on systems of differential inclusions with one-dimentional ϕ-Laplacian.

NonMonotonicity of Traveling Wave Profiles for a Unimodal Recursive System

Recursive systems of unimodal type may exhibit rich propagation dynamics. The shape of traveling wave profiles is one of the unsolved questions in this challenging topic. In this paper, we obtain criteria for the nonmonotonicity of wave profiles in terms of an eigenvalue problem. The proofs rely on establishing several nonlocal Harnack type inequalities, which may be of interest on their own. The existence and uniqueness of traveling waves are also obtained based on these inequalities.

A matrix Harnack inequality for semilinear heat equations

<abstract><p>We derive a matrix version of Li & Yau–type estimates for positive solutions of semilinear heat equations on Riemannian manifolds with nonnegative sectional curvatures and parallel Ricci tensor, similarly to what R. Hamilton did in <sup>[<xref ref-type="bibr" rid="b5">5</xref>]</sup> for the standard heat equation. We then apply these estimates to obtain some Harnack–type inequalities, which give local bounds on the solutions in terms of the geometric quantities involved.</p></abstract>

On the localization and numerical computation of positive radial solutions for ϕ -Laplace equations in the annulus

The paper deals with the existence and localization of positive radial solutions for stationary partial differential equations involving a general ϕ -Laplace operator in the annulus. Three sets of boundary conditions are considered: Dirichlet–Neumann, Neumann–Dirichlet and Dirichlet–Dirichlet. The results are based on the homotopy version of Krasnosel'skii's fixed point theorem and Harnack type inequalities, first established for each one of the boundary conditions. As a consequence, the problem of multiple solutions is solved in a natural way. Numerical experiments confirming the theory, one for each of the three sets of boundary conditions, are performed by using the MATLAB object-oriented package Chebfun.

A Note on Harnack Type Inequality for the Gaussian Curvature Flow

A Note on Harnack Type Inequality for the Gaussian Curvature Flow

Global Regularity Results for Non-homogeneous Growth Fractional Problems

This article concerns with the global H\"older regularity of weak solutions to a class of problems involving the fractional $(p,q)$-Laplacian, denoted by $(-\Delta)^{s_1}_{p}+(-\Delta)^{s_2}_{q}$, for $1<p,q<\infty$ and $s_1,s_2\in (0,1)$. We use a suitable Caccioppoli inequality and local boundedness result in order to prove the weak Harnack type inequality. Consequently, by employing a suitable iteration process, we establish the interior H\"older regularity for local weak solutions, which need not be assumed bounded. The global H\"older regularity result we prove expands and improves the regularity results of Giacomoni, Kumar and Sreenadh (arXiv: 2102.06080) to the subquadratic case (that is, $q<2$) and more general right hand side, which requires a different and new approach. Moreover, we establish a nonlocal Harnack type inequality for weak solutions, which is of independent interest.

Maximum principle for higher order operators in general domains

Abstract We first prove De Giorgi type level estimates for functions in W 1,t (Ω), Ω ⊂ R N $ \Omega\subset{\mathbb R}^N $ , with t &gt; N ≥ 2 $ t \gt N\geq 2 $ . This augmented integrability enables us to establish a new Harnack type inequality for functions which do not necessarily belong to De Giorgi’s classes as obtained in Di Benedetto–Trudinger [10] for functions in W 1,2(Ω). As a consequence, we prove the validity of the strong maximum principle for uniformly elliptic operators of any even order, in fairly general domains in dimension two and three, provided second order derivatives are taken into account.

Upper bounds of Hessian matrix and gradient estimates of positive solutions to the nonlinear parabolic equation along Ricci flow

In the paper, we first prove a Hamilton-Souplet-Zhang type gradient estimate for a positive solution to the nonlinear parabolic type equation ∂tu(x,t)=(Δ−q(x,t))u(x,t)+au(x,t)logu(x,t)on Riemaniann manifolds along the Ricci flow. These estimates optimize the obtained conclusions by Bailesteanu et al. (2010) and Li and Zhu (2018). Secondly, by using Han-Zhang’s method (Han and Zhang, 2016) and Hamilton-Souplet-Zhang type gradient estimate, we establish some global and local upper bounds for the Hessian of log positive solutions of the nonlinear parabolic type equations along the Ricci flow. As an application, we deduce some space-only Harnack type inequalities for bounded positive solutions of the parabolic type equation. Once more, by using Li’s method (Li, 1991), we also derive some second order gradient estimates. Finally, we derive L2 estimates for log-solutions to the parabolic type equation on Riemaniann manifolds.

Asymptotic estimates for an integral equation in theory of phase transition

In this paper, we study the asymptotic behavior of solutions of an integral equation of the Allen–Cahn type in , when |x| → ∞. Here is uniformly continuous, and k ⩾ 1, n ⩾ 2, α ∈ (0, n) and . In addition, is a constant vector and C * is a real constant. If for some s ∈ [1, ∞), we know that |u| → 1 when |x| → ∞. Furthermore, we prove that if for some , then when |x| → ∞, and hence . When for some , then there exists some positive constant C such that |1 − |u(x)|2| ⩽ C|x| α−n for large |x|. Here the Harnack type estimate and the regularity lifting lemma come into play in those proofs.

Harnack type inequalities for operators in logarithmic submajorisation

The aim of this paper is to study the Harnack type logarithmic submajorisation and Fuglede-Kadison determinant inequalities for operators in a finite von Neumann algebra. In particular, the Harnack type determinant inequalities due to Lin-Zhang [15] and Yang-Zhang [27] are extended to the case of operators in a finite von Neumann algebra.

Path-distribution dependent SDEs with singular coefficients

In this paper, existence and uniqueness are proved for path-dependent McKean-Vlasov type SDEs with integrability conditions. Gradient estimates and the Harnack type inequalities are derived in the case that the drifts are Dini continuous in the space variable. These generalize the corresponding results derived for classical functional SDEs with singular coefficients.

Quasi-Harnack inequality

In this paper we obtain some extensions of the classical Krylov-Safonov Harnack inequality. The novelty is that we consider functions that do not necessarily satisfy an infinitesimal equation but rather exhibit a two-scale behavior. We require that at scale larger than some $r_0>0$ (small) the functions satisfy the comparison principle with a standard family of quadratic polynomials, while at scale $r_0$ they satisfy a Weak Harnack type estimate.

Harnack Type Inequalities and Multiple Solutions in Cones of Nonlinear Problems

The paper presents an abstract theory regarding operator equations and systems in ordered Banach spaces. We obtain existence, localization and multiplicity results of positive solutions using Krasnosel'skii's fixed point theorem in cones, and a Harnack type inequality. Concerning systems, the localization is established by the vector version of Krasnosel'skii's theorem, where the compression-expansion conditions are expressed on components. The approach is sufficiently general to cover and unify a large number of results on particular classes of problems. It also can guide future research in this direction.