Narrow Results

Inspired by persistent homology in topological data analysis, we introduce the homological eigenvalues of the graph $p$-Laplacian ${\mathrm{\Delta }}_p$, which allows us to analyze and classify non-variational eigenvalues. We show the stability of homological eigenvalues, and we prove that for any homological eigenvalue $\lambda ({\mathrm{\Delta }}_p)$, the function $p\mapsto p{(2\lambda ({\mathrm{\Delta }}_p))}^{\frac{1}{p}}$ is locally increasing, while the function $p\mapsto 2^{-p}\lambda ({\mathrm{\Delta }}_p)$ is locally decreasing. As a special class of homological eigenvalues, the min–max eigenvalues ${\lambda }_1({\mathrm{\Delta }}_p),\dots ,{\lambda }_k({\mathrm{\Delta }}_p),\dots ,$ are locally Lipschitz continuous with respect to $p\in [1,+\infty )$. We also establish the monotonicity of $p{(2{\lambda }_k({\mathrm{\Delta }}_p))}^{\frac{1}{p}}$ and $2^{-p}{\lambda }_k({\mathrm{\Delta }}_p)$ with respect to $p\in [1,+\infty )$.

These results systematically establish a refined analysis of ${\mathrm{\Delta }}_p$-eigenvalues for varying $p$, which lead to several applications, including: (1) settle an open problem by Amghibech on the monotonicity of some function involving eigenvalues of $p$-Laplacian with respect to $p$; (2) resolve a question asking whether the third eigenvalue of graph $p$-Laplacian is of min–max form; (3) refine the higher-order Cheeger inequalities for graph $p$-Laplacians by Tudisco and Hein, and extend the multi-way Cheeger inequality by Lee, Oveis Gharan and Trevisan to the $p$-Laplacian case.

Furthermore, for the 1-Laplacian case, we characterize the homological eigenvalues and min–max eigenvalues from the perspective of topological combinatorics, where our idea is similar to the authors’ work on discrete Morse theory.

The goal of this paper is to prove new upper bounds for the first positive eigenvalue of the $p$-Laplacian operator in terms of the mean curvature and constant sectional curvature on Riemannian manifolds. In particular, we provide various estimates of the first eigenvalue of the $p$-Laplacian operator on closed orientate $n$-dimensional Lagrangian submanifolds in a complex space form ${𝕄}^n(4𝜖)$ with constant holomorphic sectional curvature $4𝜖$. As applications of our main theorem, we generalize the Reilly-inequality for the Laplacian [R. C. Reilly, On the first eigenvalue of the Laplacian for compact submanifolds of Euclidean space, Comment. Math. Helv. 52(4) (1977) 525–533] to the $p$-Laplacian for a Lagrangian submanifold in a complex Euclidean space and complex projective space for $𝜖=0$ and $𝜖=1$, respectively.

In this work, we prove optimal $W^{s,p}$-approximation estimates (with $p\in [1,+\infty ]$) for elliptic projectors on local polynomial spaces. The proof hinges on the classical Dupont–Scott approximation theory together with two novel abstract lemmas: An approximation result for bounded projectors, and an $L^p$-boundedness result for $L^2$-orthogonal projectors on polynomial subspaces. The $W^{s,p}$-approximation results have general applicability to (standard or polytopal) numerical methods based on local polynomial spaces. As an illustration, we use these $W^{s,p}$-estimates to derive novel error estimates for a Hybrid High-Order (HHO) discretisation of Leray–Lions elliptic problems whose weak formulation is classically set in $W^{1,p}(\mathrm{\Omega })$ for some $p\in (1,+\infty )$. This kind of problems appears, e.g. in the modelling of glacier motion, of incompressible turbulent flows, and in airfoil design. Denoting by $h$ the meshsize, we prove that the approximation error measured in a $W^{1,p}$-like discrete norm scales as $h^{\frac{k+1}{p-1}}$ when $p\ge 2$ and as $h^{(k+1)(p-1)}$ when $p<2$.

We analyze a Discontinuous Galerkin method for a problem with linear advection–reaction and $p$-type diffusion, with Sobolev indices $p\in (1,\infty )$. The discretization of the diffusion term is based on the full gradient including jump liftings and interior-penalty stabilization while, for the advective contribution, we consider a strengthened version of the classical upwind scheme. The developed error estimates track the dependence of the local contributions to the error on the local Péclet numbers. A set of numerical tests support the theoretical derivations.

In this paper, we study the following boundary value problem involving the weak $p$-Laplacian.

This paper studies the existence of positive solutions for a class of singular boundary value problems of p-Laplacian. By using Global Continuation Theorem and the fixed point index technique, criteria of the existence, multiplicity and nonexistence of positive solutions are established.

In this paper, we will introduce the rotation number for the one-dimensional asymmetric p-Laplacian with a pair of periodic potentials. Two applications of this notion will be given. One is a clear characterization of two unbounded sequences of Fučik curves of the periodic Fučik spectrum of the p-Laplacian with potentials. With the help of the Poincaré–Birkhoff fixed point theorem, the other application is some existence result of multiple periodic solutions of nonlinear ordinary differential equations concerning with the p-Laplacian.

We will study the dependence of λ(a, b), half-eigenvalues of the one-dimensional p-Laplacian, on potentials , 1 ≤ γ ≤ ∞, where . Two results are obtained. One is the continuity of half-eigenvalues in , where w_γ is the weak topology in space. The other is the continuous differentiability of half-eigenvalues in , where ‖ ⋅ ‖_γ is the L^γ norm of . These results will be used to study extremal problems of half-eigenvalues in future work.

In this paper, we study the existence of sign-changing solutions for the p-Laplacian equation

In this paper we will use variational methods and limiting approaches to give a complete solution to the maximization problems of the Dirichlet/Neumann eigenvalues of the one-dimensional p-Laplacian with integrable potentials of fixed L¹-norm.

We study the following boundary value problem with a concave–convex nonlinearity:

We consider a parametric nonlinear Dirichlet problem driven by the p-Laplacian, with a singular term and a p-superlinear perturbation, which need not satisfy the usual Ambrosetti–Rabinowitz condition. Using variational methods together with truncation techniques, we prove a bifurcation-type theorem describing the behavior of the set of positive solutions as the parameter varies.

To characterize the complete structure of the Fučík spectrum of the $p$-Laplacian on higher dimensional domains is a long-standing problem. In this paper, we study the $p$-Laplacian with integrable potentials on an interval under the Dirichlet or the Neumann boundary conditions. Based on the strong continuity and continuous differentiability of solutions in potentials, we will give a comprehensive characterization of the corresponding Fučík spectra: each of them is composed of two trivial lines and a double-sequence of differentiable, strictly decreasing, hyperbolic-like curves; all asymptotic lines of these spectral curves are precisely described by using eigenvalues of the $p$-Laplacian with potentials; and moreover, all these spectral curves have strong continuity in potentials, i.e. as potentials vary in the weak topology, these spectral curves are continuously dependent on potentials in a certain sense.

In this paper, by Morse theory, we study the existence and multiplicity of solutions for the $p$-Laplacian equation with a “concave” nonlinearity and a parameter. In our results, we do not need any additional global condition on the nonlinearities, except for a subcritical growth condition.

In this paper, the minimization problem

In this paper, we establish sharp $C^{1,\alpha }$ estimates for weak solutions of singular and degenerate quasilinear elliptic equation

Let $G=(V,E)$ be a connected finite graph and $C(V)$ be the set of functions defined on $V$. Let ${\mathrm{\Delta }}_p$ be the discrete $p$-Laplacian on $G$ with $p>1$ and $L={\mathrm{\Delta }}_p-k$, where $k\in C(V)$ is positive everywhere. Consider the operator $L:C(V)\to C(V)$. We prove that $-L$ is one-to-one, onto and preserves order. So it implies that there exists a unique solution to the equation $Lu=f$ for any given $f\in C(V)$. We also prove that the equation ${\mathrm{\Delta }}_{p}u=\overline{f}-f$ has a solution which is unique up to a constant, where $\overline{f}$ is the average of $f$. With the help of these results, we finally give various conditions such that the $p$th Kazdan–Warner equation ${\mathrm{\Delta }}_{p}u=c-h{e}^u$ has a solution on $V$ for given $h\in C(V)$ and $c\in ℝ$. Thus we generalize Grigor’yan, Lin and Yang’s work Kazdan–Warner equation on graph, Calc. Var. Partial Differential Equations55(4) (2016) Paper No. 92, 13 pp. for $p=2$ to any $p>1$.

We consider a nonlinear elliptic equation driven by the p-Laplacian with a discontinuous nonlinearity. Such problems have a "multivalued" and a "single-valued" interpretation. We are interested in the latter and we prove the existence of at least two distinct solutions, both smooth and one strictly positive. Our approach is variational based on the nonsmooth critical point theory for locally Lipschitz functions, coupled with penalization and truncation techniques.

In this paper, we establish the existence of non-trivial weak solutions in , 1 < p < ∞, to a class of non-uniformly elliptic equations of the form

Let Ω ⊂ ℝ^N, N ≥ 2, be a smooth bounded domain. It is shown that: (a) if and ess inf_{x ∈ y} p(x) > 1, then the generalized Lebesgue space (L^{p (·)}(Ω), ‖·‖_p(·)) is smooth; (b) if and p(x) > 1, , then the generalized Sobolev space is smooth. In both situations, the formulae for the Gâteaux gradient of the norm corresponding to each of the above spaces are given; (c) if and p(x) ≥ 2, , then is uniformly convex and smooth.