We study stability and Hölder regularity up to the boundary for the quaternionic m-Hessian equation H_m(u):=(Δ u)^m∧β ^n-m=f dV on bounded strongly m-hyperconvex domains in ℍ^n . We prove the capacity estimate Vol (K)≤ C_τCap_m(K,Ω )^τ , 1<τ <2n/2n-m, for every compact set K⋐Ω , using Sroka’s quaternionic Monge–Ampère L^q , q>2 , estimate and the Maclaurin comparison between the n- and m-Hessian operators. It implies that every density f∈ L^p , p>2n/m , is dominated by m-Hessian capacity with a positive gain. A sublevel capacity inequality and a De Giorgi iteration then yield one-sided L^1 -to- L^∞ stability and density stability. If solvable data have uniformly bounded L^p densities converging in L^1 and boundary values converging uniformly, the corresponding solutions converge uniformly to a unique continuous weak solution with the limiting Hessian measure and boundary value. Assuming smooth solvability for strictly positive smooth data, we obtain continuous solvability for arbitrary nonnegative L^p densities and continuous boundary data. If the boundary datum and a subsolution are C^0,a , then the solution is Hölder continuous on Ω with every exponent 0<γ
Abstract This paper studies the Alexandrov–Fenchel inequality $$\int _{\Sigma } H_{n-2} d A \ge \omega _n^{\frac{n-2}{n-1}} A(\Sigma )^{\frac{1}{n-1}}$$ ∫ Σ H n - 2 d A ≥ ω n n - 2 n - 1 A ( Σ ) 1 n - 1 for $$C^{1,1}$$ C 1 , 1 hypersurfaces $$\Sigma \subset \mathbb {R}^n$$ Σ ⊂ R n , where $$\omega _n$$ ω n is the surface area of $$\mathbb {S}^{n-1}$$ S n - 1 . We prove this inequality for axiconvex hypersurfaces and extend it to axisymmetric cases. For any axisymmetric $$\Sigma $$ Σ , we derive an asymptotic estimate for $$\frac{1}{\operatorname {diam}(\Sigma )} \int _{\Sigma } H_{n-2} d A-\frac{\omega _{n-1}}{n-1}$$ 1 diam ( Σ ) ∫ Σ H n - 2 d A - ω n - 1 n - 1 . Moreover, we prove a stability estimate for the mean width-diameter inequality of convex bodies.
Abstract In this paper, we investigate the existence and multiplicity of solutions for an anisotropic variable exponent problem of the form $$\begin{aligned} -\sum _{i=1}^{N}\frac{\partial }{\partial x_{i}} \left( \left| \frac{\partial u}{\partial x_{i}}\right| ^{p_{i}(x)-2} \frac{\partial u}{\partial x_{i}}\right) +\sum _{i=1}^{N}| u|^{p_{i}(x)-2}u = \lambda k(x)| u|^{\alpha (x)-2}u \end{aligned}$$ - ∑ i = 1 N ∂ ∂ x i ∂ u ∂ x i p i ( x ) - 2 ∂ u ∂ x i + ∑ i = 1 N | u | p i ( x ) - 2 u = λ k ( x ) | u | α ( x ) - 2 u in $$\Omega $$ Ω , subject to the anisotropic Neumann boundary condition $$\begin{aligned} \sum _{i=1}^{N}\left| \frac{\partial u}{\partial x_{i}}\right| ^{p_{i}(x)-2} \frac{\partial u}{\partial x_{i}}\eta _{i} =h(x)| u|^{\beta (x)-2}u \quad {\text {on }} \partial \Omega , \end{aligned}$$ ∑ i = 1 N ∂ u ∂ x i p i ( x ) - 2 ∂ u ∂ x i η i = h ( x ) | u | β ( x ) - 2 u on ∂ Ω , where $$\Omega \subset \mathbb {R}^N$$ Ω ⊂ R N , $$N\ge 3$$ N ≥ 3 , is a bounded domain with smooth boundary and $$\begin{aligned} \Delta _{\vec {p}(\cdot )} u= \sum _{i=1}^N \dfrac{\partial }{\partial x_i}\left( \left| \dfrac{\partial u}{\partial x_i}\right| ^{p_i(x)-2} \dfrac{\partial u}{\partial x_i}\right) \end{aligned}$$ Δ p → ( · ) u = ∑ i = 1 N ∂ ∂ x i ∂ u ∂ x i p i ( x ) - 2 ∂ u ∂ x i denotes the anisotropic variable exponent $$\vec {p}(\cdot )$$ p → ( · ) -Laplace differential operator. Under suitable assumptions on the data, we prove that the above problem admits at least two nontrivial nonnegative weak solutions. The proof is based on the Nehari manifold approach combined with the fibering map technique and the direct method of the calculus of variations.
We prove that there admits no nontrivial L^p (p>1) harmonic 1-forms on a complete noncompact hypersurface Σ ^n in a product manifold 𝕊^n(√(2(n-1)))×ℝ (n≥ 3) if the square of L^n -norm of the traceless second fundamental form is less than min{pα ^2,2p^-1n^2}/C_0pn(n-1)(n^2-2n+2) or the square of the length of the traceless second fundamental form is less than nα ^2/(n-1)(n^2-2n+2) , where α is an angle function and C_0 is the Sobolev constant depending only on n. These are extensions of results on hypersurfaces in spheres and complex projective spaces. They are also generalization of results on hypersurfaces in 𝕊^n×ℝ from L^2 -harmonic 1-forms to L^p case.
We show that the Kähler–Einstein metrics on the four families of symmetric toric Fano manifolds due to Batyrev and Selivanova cannot be induced by immersions into projective spaces equipped with the Fubini–Study metric. We also show a similar conclusion for the non-symmetric examples discovered by Nill and Paffenholz. As an application we obtain that a centrally symmetric toric Fano manifold admits a Kähler–Einstein metric induced by a projective immersion if and only if it is a product of projective lines. These results confirm for each of these examples a broader conjecture characterizing the Kähler–Einstein metrics that can be induced by projective immersions.
