We construct the L 2-gradient flow of the ideal functional, which is defined as the squared integral of the derivative of curvature, under the local length constraint. We prove that (i) the Cauchy problem of the gradient flow has a unique global-in-time solution, and (ii) the global-in-time solution converges to a critical point of the ideal functional under the total length constraint. As a corollary, we show the existence of a critical point with a rotation number of zero.

We consider the obstacle problem under Dirichlet boundary conditions for Euler's elastica functional in the class of one-dimensional symmetric graphs over the real axis. We prove the optimality of a previously obtained smallness condition on the size of obstacles such that the obstacle problem possesses a minimiser.

In this article we study the $H^1(du)$-gradient flow for the energy $E[X] = Q[X]/A[X]$ where $Q[X]$ is the Dirichlet energy of $X$, $A[X]$ is the signedenclosed area of $X$, and $X:\mathbb{S}\rightarrow\mathbb{R}^2$ is a $H^1(du)$ map. We prove that solutions with initially positive signed enclosed area exist eternally, and converge as $t\rightarrow\infty$ to a (possibly multiply-covered) circle. In this way we recover a parametrised isoperimetric inequality for $H^1(du)$ maps.

In this paper we study the H2(ds)-gradient flow for the modified elastic energy defined on closed immersed curves in Rn. We prove the existence of a unique global-in-time solution to the flow and establish full convergence to elastica by way of a Lojasiewicz-Simon gradient inequality.

We consider the curve diffusion flow for open planar curves with boundary on two skew lines. For each angle θ∈ (0, π ) of the two skew lines, we prove the existence of global-in-time solutions under a suitable initial condition. Furthermore, we show the full limit convergence of solutions to the arc of the sector with the central angle θ and the same area as that of the initial curve.

In this paper we consider an obstacle problem for a generalization of the p-elastic energy among graphical curves with fixed ends. Taking into account that the Euler–Lagrange equation has a degeneracy, we address the question whether solutions have a flat part, i.e. an open interval where the curvature vanishes. We also investigate which is the main cause of the loss of regularity, the obstacle or the degeneracy. Moreover, we give several conditions on the obstacle that assure existence and nonexistence of solutions. The analysis can be refined in the special case of the p-elastica functional, where we obtain sharp existence results and uniqueness for symmetric minimizers.

We consider obstacle problems for the Willmore functional in the class of graphs of functions and surfaces of revolution with Dirichlet boundary conditions. We prove the existence of minimisers of the obstacle problems under the assumption that the Willmore energy with the unilateral constraint is below a universal bound. We address the question whether such bounds are necessary in order to ensure the solvability of the obstacle problems. Moreover, we give several instructive examples of obstacles such that minimisers exist.

Abstract In our previous paper [K. Ishige, S. Okabe, and T. Sato, A supercritical scalar field equation with a forcing term, J. Math. Pures Appl. 128 (2019), pp. 183–212], we proved the existence of a threshold κ ∗ > 0 {\kappa }^{\ast }\gt 0 such that the elliptic problem for an inhomogeneous elliptic equation − Δ u + u = u p + κ μ -\Delta u+u={u}^{p}+\kappa \mu in R N {{\bf{R}}}^{N} possesses a positive minimal solution decaying at the space infinity if and only if 0 < κ ≤ κ ∗ 0\lt \kappa \le {\kappa }^{\ast } . Here, N ≥ 2 N\ge 2 , μ \mu is a nontrivial nonnegative Radon measure in R N {{\bf{R}}}^{N} with a compact support, and p > 1 p\gt 1 is in the Joseph-Lundgren subcritical case. In this article, we prove the existence of nonminimal positive solutions to the elliptic problem. Our arguments are also applicable to inhomogeneous semilinear elliptic equations with exponential nonlinearity.

In this paper, we consider the L2-gradient flow for the modified p-elastic energy defined on planar closed curves. We formulate a notion of weak solution for the flow and prove the existence of global-in-time weak solutions with p≥2 for initial curves in the energy space via minimizing movements. Moreover, we prove the existence of unique global-in-time solutions to the flow with p=2 and obtain their subconvergence to an elastica as t→∞.

Abstract In this paper we study the existence and the nonexistence of solutions to an inhomogeneous non-linear elliptic problem (P) −Δu+u=F(u)+κμ in RN, u>0 in RN, u(x)→0 as |x|→∞, - \Delta u + u = F(u) + \kappa \mu \quad {\kern 1pt} {\rm in}{\kern 1pt} \quad {{\bf R}^N},\quad u > 0\quad {\kern 1pt} {\rm in}{\kern 1pt} \quad {{\bf R}^N},\quad u(x) \to 0\quad {\kern 1pt} {\rm as}{\kern 1pt} \quad |x| \to \infty , where F = F(t) grows up (at least) exponentially as t → ∞. Here N ≥ 2, κ > 0, and μ∈Lc1(RN)\{0} \mu \in L_{\rm{c}}^1({{\bf R}^N})\backslash \{ 0\} is nonnegative. Then, under a suitable integrability condition on μ, there exists a threshold parameter κ* > 0 such that problem (P) possesses a solution if 0 < κ < κ* and it does not possess no solutions if κ > κ*. Furthermore, in the case of 2 ≤ N ≤ 9, problem (P) possesses a unique solution if κ = κ*.

We establish the existence of solutions to the Cauchy problem for a large class of nonlinear parabolic equations including fractional semilinear parabolic equations, higher-order semilinear parabolic equations, and viscous Hamilton-Jacobi equations by using the majorant kernel introduced in Ishige et al. (2020). (C) 2022 Elsevier Ltd. All rights reserved.

In this paper we consider the initial-boundary value problem for a fourth order parabolic equation with gradient nonlinearity. The problem is regarded as the L2-gradient flow for an energy functional which is unbounded from below. We first prove the existence and the uniqueness of solutions to the problem via the Galerkin method. Moreover, combining the potential well method with the Galerkin method, we study the asymptotic behavior of global-in-time solutions to the problem.

This paper is concerned with the positivity of solutions to the Cauchy problem for linear and nonlinear parabolic equations with the biharmonic operator as fourth order elliptic principal part. Generally, Cauchy problems for parabolic equations of fourth order have no positivity preserving property due to the change of sign of the fundamental solution. One has eventual local positivity for positive initial data, but on short time scales, one will in general have also regions of negativity. The first goal of this paper is to find sufficient conditions on initial data which ensure the existence of solutions to the Cauchy problem for the linear biharmonic heat equation which are positive for all times and in the whole space. The second goal is to apply these results to show existence of globally positive solutions to the Cauchy problem for a semilinear biharmonic parabolic equation.

We establish the existence of solutions of the Cauchy problem for a higher-order semilinear parabolic equation by introducing a new majorizing kernel. We also study necessary conditions on the initial data for the existence of local-in-time solutions and identify the strongest singularity of the initial data for the solvability of the Cauchy problem.

This paper is concerned with the obstacle problem for the L2-gradient flow for a functional which is higher order, non-convex and unbounded from below. We prove (i) the existence and uniqueness of local-in-time solutions to the obstacle problem and (ii) a gradient structure of the functional of the solutions, via minimizing movements. Moreover, we show the existence of solutions which blow up in a finite time.

AbstractWe consider the variational inequality on modified elastic graphs. Since the variational inequality is derived from the minimization problem for the modified elastic energy defined on graphs with the unilateral constraint, a solution to the variational inequality can be constructed by the direct method of calculus of variations. In this paper we prove the existence of solutions to the variational inequality via a dynamical approach. More precisely, we construct an L2-type gradient flow corresponding to the variational inequality and prove the existence of solutions to the variational inequality via the study on the limit of the flow.

Let $u$ be a solution to the Cauchy problem for a fourth-order nonlinear parabolic equation $\partial_t u+(-\Delta)^2u=-\nabla\cdot(|\nabla u|^{p-2}\nabla u)$ on ${\bf R}^N$, where $p>2$ and $N\ge 1$. In this paper we give a sufficient condition for the maximal existence time $T_M(u)$ of the solution $u$ to be finite. Furthermore, we show that if $T_M(u)<\infty$, then $\|\nabla u(t)\|_{L^\infty({\bf R}^N)}$ blows up at $t=T_M(u)$, and we obtain lower estimates on the blow-up rate. We also give a sufficient condition on the existence of global-in-time solutions to the Cauchy problem.

We consider a fourth order nonlinear eigenvalue problem with an adhesive constraint. The problem is regarded as a generalization of the buckling eigenvalue problem with the clamped boundary condition. We prove the existence of the first eigenvalue of the problem and show that the corresponding eigenfunction does not have "flat core of adhesion type".

In this paper we establish a general form of the isoperimetric inequality for immersed closed curves (possibly non-convex) in the plane under rotational symmetry. As an application, we obtain a global existence result for the surface diffusion flow, providing that an initial curve is H^2 -close to a multiply covered circle and is sufficiently rotationally symmetric.

This paper is concerned with the elliptic problem for a scalar field equation with a forcing term \begin{equation} \tag{P}-\Delta u+u=u^p+ \kappa \mu \quad \mbox{in} \quad{\bf R}^N, \quad u>0 \quad \mbox{in} \quad {\bf R}^N, \quad u(x)\to 0\quad \mbox{as} \quad |x| \to \infty, \end{equation} where $N\ge 2$, $p>1$, $\kappa>0$ and $\mu$ is a Radon measure in ${\bf R}^N$ with a compact support. Under a suitable integrability condition on $\mu$, we give a complete classification of the solvability of problem~(P) with $1