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For Laplacians defined by measures on a bounded domain in ℝn, we prove analogues of the classical eigenvalue estimates for the standard Laplacian: lower bound of sums of eigenvalues by Li and Yau, and gaps of consecutive eigenvalues by Payne, Pólya and Weinberger. This work is motivated by the study of spectral gaps for Laplacians on fractals.
where b > 0, p ∈ (4, 6), the potential
$V\in C(\mathbb R^3,\mathbb R)$
and ɛ is a positive parameter. The existence and multiplicity of semi-classical state solutions are obtained by variational method for this problem with several classes of critical frequency potentials, i.e.,
$\inf _{\mathbb R^N} V=0$
. As to Kirchhoff type problem, little has been done for the critical frequency cases in the literature, especially the potential may vanish at infinity.
We show that the spectrum of the relativistic mean curvature operator on a bounded domain Ω ⊂ ℝN (N ⩾ 1) having smooth boundary, subject to the homogeneous Dirichlet boundary condition, is exactly the interval (λ1(2), ∞), where λ1(2) stands for the principal frequency of the Laplace operator in Ω.
Within the framework of the generalised Landau-de Gennes theory, we identify a Q-tensor-based energy that reduces to the four-constant Oseen–Frank energy when it is considered over orientable uniaxial nematic states. Although the commonly considered version of the Landau-de Gennes theory has an elastic contribution that is at most cubic in components of the Q-tensor and their derivatives, the alternative offered here is quartic in these variables. One clear advantage of our approach over the cubic theory is that the associated minimisation problem is well-posed for a significantly wider choice of elastic constants. In particular, this quartic energy can be used to model nematic-to-isotropic phase transitions for highly disparate elastic constants. In addition to proving well-posedness of the proposed version of the Landau-de Gennes theory, we establish a rigorous connection between this theory and its Oseen–Frank counterpart via a Г-convergence argument in the limit of vanishing nematic correlation length. We also prove strong convergence of the associated minimisers.
where 0 < γ < 1, λ > 0 and 0 < s ≤ t < 1 with 4s + 2t > 3. Under certain assumptions on V and f, we show the existence, uniqueness, and monotonicity of positive solution uλ using the variational method. We also give a convergence property of uλ as λ → 0, when λ is regarded as a positive parameter.
We revisit the problem of approximating minimizers of certain convex functionals subject to a convexity constraint by solutions of fourth order equations of Abreu type. This approximation problem was studied in previous articles of Carlier–Radice (Approximation of variational problems with a convexity constraint by PDEs of Abreu type. Calc. Var. Partial Differential Equations58 (2019), no. 5, Art. 170) and the author (Singular Abreu equations and minimizers of convex functionals with a convexity constraint, arXiv:1811.02355v3, Comm. Pure Appl. Math., to appear), under the uniform convexity of both the Lagrangian and constraint barrier. By introducing a new approximating scheme, we completely remove the uniform convexity of both the Lagrangian and constraint barrier. Our analysis is applicable to variational problems motivated by the original 2D Rochet–Choné model in the monopolist's problem in Economics, and variational problems arising in the analysis of wrinkling patterns in floating elastic shells in Elasticity.
We consider the fractional critical problem
$A_{s}u=K(x)u^{(n+2s)/(n-2s)},u>0$
in
$\unicode[STIX]{x1D6FA},u=0$
on
$\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$
, where
$A_{s},s\in (0,1)$
, is the fractional Laplace operator and
$K$
is a given function on a bounded domain
$\unicode[STIX]{x1D6FA}$
of
$\mathbb{R}^{n},n\geq 2$
. This is based on A. Bahri’s theory of critical points at infinity in Bahri [Critical Points at Infinity in Some Variational Problems, Pitman Research Notes in Mathematics Series, 182 (Longman Scientific & Technical, Harlow, 1989)]. We prove Bahri’s estimates in the fractional setting and we provide existence theorems for the problem when
$K$
is close to 1.
where N ⩾ 2, Iα is the Riesz potential with order α ∈ (0, N − 1) and 2 ⩽ p < (N + α)/(N − 2). When the potential V is assumed to be bounded and bounded away from zero, we construct a family of localized bound states of higher topological type that concentrate around the local minimum points of the potential V as ε → 0. These solutions are obtained by combining the Byeon–Wang's penalization approach and the classical symmetric mountain pass theorem.
We discuss the existence, nonexistence and multiplicity of solutions for a class of elliptic equations in the unit ball with zero Dirichlet boundary conditions involving nonlinearities with supercritical growth. By using Pohozaev type identity we prove a nonexistence result for a class of supercritical problems with variable exponent which allow us to complement the analysis developed in (Calc. Var. (2016) 55:83). Moreover, we establish existence results of positive solutions for semilinear elliptic equations involving nonlinearities which are subcritical at infinity just in a part of the domain, and can be supercritical in a suitable sense.
We prove that if two
$C^{1,1}(\unicode[STIX]{x1D6FA})$
solutions of the second boundary value problem for the generated Jacobian equation intersect in
$\unicode[STIX]{x1D6FA}$
then they are the same solution. In addition, we extend this result to
$C^{2}(\overline{\unicode[STIX]{x1D6FA}})$
solutions intersecting on the boundary, via an additional convexity condition on the target domain.
We consider the problem of minimizing the weighted Dirichlet energy between homeomorphisms of planar annuli. A known challenge lies in the case when the weight λ depends on the independent variable z. We prove that for an increasing radial weight λ(z) the infimal energy within the class of all Sobolev homeomorphisms is the same as in the class of radially symmetric maps. For a general radial weight λ(z) we establish the same result in the case when the target is conformally thin compared to the domain. Fixing the admissible homeomorphisms on the outer boundary we establish the radial symmetry for every such weight.
where σD is a modulus of continuity defined in the interval [0, |D|] and depends only on the domain D. The inequality is optimal for any domain D and for any values of
$\|f\|_1$
and
$\|f\|_\infty .$
We also show that
\sigma_D(t)\leq\sigma_B(t),\text{ for }t\in[0,|D|],
where B is a ball and |B| = |D|. Using this optimality property of σD, we derive Brezis–Galloute–Wainger type inequalities on the L∞ norm of u in terms of the L1 and L∞ norms of f. As an application we derive L∞ − L1 estimates on the k-th Laplace eigenfunction of the domain D.
where λ > 0 is a real parameter, f belongs to a suitable Lebesgue space,
$\mu \in L^{\infty}$
and
$\mathbb {D}_s^2$
is a nonlocal ‘gradient square’ term given by
Depending on the real parameter λ > 0, we derive existence and non-existence results. The proof of our existence result relies on sharp Calderón–Zygmund type regularity results for the fractional Poisson equation with low integrability data. We also obtain existence results for related problems involving different nonlocal diffusion terms.
where
$\unicode[STIX]{x1D6FA}$
is a smooth bounded domain in
$\mathbb{R}^{N}$
,
$N\geq 2$
. This work generalizes the well-known works on the Lyapunov inequality for extremal Pucci’s equations with gradient nonlinearity.
We consider the minimizing problem for the energy functional with prescribed mass constraint related to the fractional non-linear Schrödinger equation with periodic potentials. Using the concentration-compactness principle, we show a complete classification for the existence and non-existence of minimizers for the problem. In the mass-critical case, under a suitable assumption of the potential, we give a detailed description of blow-up behaviour of minimizers once the mass tends to a critical value.
where
$\beta \in {\mathbb R}$
is a coupling constant,
$\mu ,\nu $
are positive constants, P,Q are weight functions decaying exponentially to zero at infinity, α can be regarded as a parameter. This type of system arises, in particular, in models in Bose–Einstein condensates theory and Kerr-like photo refractive media.
We prove that, for any positive integer k > 1, there exists a suitable range of α such that the above problem has a non-radial positive solution with exactly k maximum points which tend to infinity as
$\alpha \to +\infty $
(or
$0^+$
). Moreover, we also construct prescribed number of sign-changing solutions.
In this paper we study a general eigenvalue problem for the so called (p, 2)-Laplace operator on a smooth bounded domain Ω ⊂ ℝN under a nonlinear Steklov type boundary condition, namely
For positive weight functions a and b satisfying appropriate integrability and boundedness assumptions, we show that, for all p>1, the eigenvalue set consists of an isolated null eigenvalue plus a continuous family of eigenvalues located away from zero.
where
$\unicode[STIX]{x1D6FA}\subset \mathbb{R}^{N}(N\geqslant 1)$
is a bounded domain with smooth boundary
$\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$
,
$h\in L^{1}(\unicode[STIX]{x1D6FA})$
,
$h>0$
almost everywhere in
$\unicode[STIX]{x1D6FA}$
,
$k\in L^{\infty }(\unicode[STIX]{x1D6FA})$
is a non-negative function,
$B:[0,+\infty )\rightarrow [m,+\infty )$
is continuous for some positive constant
$m$
,
$p>1$
,
$0\leqslant q\leqslant p-1$
, and
$\unicode[STIX]{x1D6FE}>1$
. A “compatibility condition” on the couple
$(h(x),\unicode[STIX]{x1D6FE})$
will be given for the problem to admit at least one solution. To be a little more precise, it is shown that the problem admits at least one solution if and only if there exists a
$u_{0}\in W_{0}^{1,p}(\unicode[STIX]{x1D6FA})$
such that
$\int _{\unicode[STIX]{x1D6FA}}h(x)u_{0}^{1-\unicode[STIX]{x1D6FE}}\text{d}x<\infty$
. When
$k(x)\equiv 0$
, the weak solution is unique.