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We treat the exterior Dirichlet problem for a class of fully nonlinear elliptic equations of the form f (λ (D 2 u)) = g (x) with prescribed asymptotic behavior at infinity. The equations of this type had been studied extensively by Caffarelli et al. [The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of the eigenvalues of the Hessian, Acta Math. 155 (1985) 261–301], Trudinger [On the Dirichlet problem for Hessian equations, Acta Math. 175 (1995) 151–164] and many others, and there had been significant discussions on the solvability of the classical Dirichlet problem via the continuity method, under the assumption that f is a concave function. In this paper, based on Perron's method, we establish an exterior existence and uniqueness result for viscosity solutions of the equations, by assuming f to satisfy certain structure conditions as in [L. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of the eigenvalues of the Hessian, Acta Math. 155 (1985) 261–301; N. S. Trudinger, On the Dirichlet problem for Hessian equations, Acta Math. 175 (1995) 151–164] but without requiring the concavity of f. The equations in our setting may embrace the well-known Monge–Ampère equations, Hessian equations and Hessian quotient equations as special cases. [ABSTRACT FROM AUTHOR]

The aim of this paper is to present the global estimate for gradient of renormalized solutions to the following quasilinear elliptic problem: − div (A (x , ∇ u)) = μ in Ω , u = 0 on ∂ Ω , in Lorentz–Morrey spaces, where Ω ⊂ ℝ n (n ≥ 2), μ is a finite Radon measure, A is a monotone Carathéodory vector-valued function defined on W 0 1 , p (Ω) and the p -capacity uniform thickness condition is imposed on the complement of our domain Ω. It is remarkable that the local gradient estimates have been proved first by Mingione in [Gradient estimates below the duality exponent, Math. Ann.346 (2010) 571–627] at least for the case 2 ≤ p ≤ n , where the idea for extending such result to global ones was also proposed in the same paper. Later, the global Lorentz–Morrey and Morrey regularities were obtained by Phuc in [Morrey global bounds and quasilinear Riccati type equations below the natural exponent, J. Math. Pures Appl.102 (2014) 99–123] for regular case p > 2 − 1 n . Here in this study, we particularly restrict ourselves to the singular case 3 n − 2 2 n − 1 < p ≤ 2 − 1 n . The results are central to generalize our technique of good- λ type bounds in the previous work [M.-P. Tran, Good- λ type bounds of quasilinear elliptic equations for the singular case, Nonlinear Anal.178 (2019) 266–281], where the local gradient estimates of solution to this type of equation were obtained in the Lorentz spaces. Moreover, the proofs of most results in this paper are formulated globally up to the boundary results. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the following Dirichlet problem for nonlinear equations involving the fractional p -Laplacian: (− Δ) p α u = f (x , u , ∇ u) , u > 0 in  Ω , u ≡ 0 in  ℝ n ∖ Ω , where Ω is a bounded or an unbounded domain which is convex in x 1 -direction, and (− Δ) p α is the fractional p -Laplacian operator defined by (− Δ) p α u (x) = C n , α , p P. V. ∫ ℝ n | u (x) − u (y) | p − 2 [ u (x) − u (y) ] | x − y | n + α p d y. Under some mild assumptions on the nonlinearity f (x , u , ∇ u) , we establish the monotonicity and symmetry of positive solutions to the nonlinear equations involving the fractional p -Laplacian in both bounded and unbounded domains. Our results are extensions of Chen and Li [Maximum principles for the fractional p-Laplacian and symmetry of solutions, Adv. Math.  335 (2018) 735–758] and Cheng et al. [The maximum principles for fractional Laplacian equations and their applications, Commun. Contemp. Math.  19(6) (2017) 1750018]. [ABSTRACT FROM AUTHOR]

Dispersive SQG equation have been studied by many works (see, e.g., [M. Cannone, C. Miao and L. Xue, Global regularity for the supercritical dissipative quasi-geostrophic equation with large dispersive forcing, Proc. Londen. Math. Soc. 106 (2013) 650–674; T. M. Elgindi and K. Widmayer, Sharp decay estimates for an anisotropic linear semigroup and applications to the surface quasi-geostrophic and inviscid Boussinesq systems, SIAM J. Math. Anal. 47 (2015) 4672–4684; A. Kiselev and F. Nazarov, Global regularity for the critical dispersive dissipative surface quasi-geostrophic equation, Nonlinearity23 (2010) 549–554; R. Wan and J. Chen, Global well-posedness for the 2D dispersive SQG equation and inviscid Boussinesq equations, Z. Angew. Math. Phys.67 (2016) 104]), which is very similar to the 3D rotating Euler or Navier–Stokes equations. Long time stability for the dispersive SQG equation without dissipation was obtained by Elgindi–Widmayer [Sharp decay estimates for an anisotropic linear semigroup and applications to the surface quasi-geostrophic and inviscid Boussinesq systems, SIAM J. Math. Anal.47 (2015) 4672–4684], where the initial condition 𝜃 0 ∈ W 3 + μ , 1 (μ > 0) plays a important role in their proof. In this paper, by using the Strichartz estimate, we can remove this initial condition. Namely, we only assume the initial data is in the Sobolev space like H s . As an application, we can also obtain similar result for the 2D Boussinesq equations with the initial data near a nontrivial equilibrium. [ABSTRACT FROM AUTHOR]

Let n ≥ 3 , Ω be a bounded (semi-)convex domain in ℝ n and the non-negative potential V belong to the reverse Hölder class RH n (ℝ n). Assume that p ∈ (1 , ∞) and ω ∈ A p (∂ Ω) , where A p (∂ Ω) denotes the Muckenhoupt weight class on ∂ Ω , the boundary of Ω. In this paper, the authors show that, for any p ∈ (1 , ∞) , the Neumann problem for the Schrödinger equation − Δ u + V u = 0 in Ω with boundary data in (weighted) L p is uniquely solvable. The obtained results in this paper essentially improve the known results which are special cases of the results obtained by Shen [Indiana Univ. Math. J.43 (1994) 143–176] and Tao and Wang [Canad. J. Math.56 (2004) 655–672], via extending the range p ∈ (1 , 2 ] of p into p ∈ (1 , ∞). [ABSTRACT FROM AUTHOR]

The main goal of this paper is to reveal the symplectic structure related to renormalization of circle maps with breaks. We first show that iterated renormalizations of r circle diffeomorphisms with d breaks, r > 2 , with given size of breaks, converge to an invariant family of piecewise Möbius maps, of dimension 2 d. We prove that this invariant family identifies with a relative character variety χ (π 1 Σ , PSL (2 , ℝ) , h) where Σ is a d -holed torus, and that the renormalization operator identifies with a sub-action of the mapping class group MCG (Σ). This action allows us to introduce the symplectic form which is preserved by renormalization. The invariant symplectic form is related to the symplectic form described by Guruprasad et al. [Group systems, groupoids, and moduli spaces of parabolic bundles, Duke Math. J.89(2) (1997) 377–412], and goes back to the earlier work by Goldman [The symplectic nature of fundamental groups of surfaces, Adv. Math.54(2) (1984) 200–225]. To the best of our knowledge the connection between renormalization in the nonlinear setting and symplectic dynamics had not been brought to light yet. [ABSTRACT FROM AUTHOR]

Given any symplectomorphism on D 2 n (n ≥ 1) which is C ∞ close to the identity, and any completely integrable Hamiltonian system Φ H t in the proper dimension, we construct a C ∞ perturbation of H such that the resulting Hamiltonian flow contains a "local Poincaré section" that "realizes" the symplectomorphism. As a (motivating) application, we show that there are arbitrarily small perturbations of any completely integrable Hamiltonian system which are entropy non-expansive (and, in particular, exhibit hyperbolic behavior on a set of positive measure). We use some results in Berger–Turaev [On Herman's positive entropy conjecture, Adv. Math. 349 (2019) 1234—1288], though in higher dimensions we could simply apply a construction from [D. Burago and S. Ivanov, Boundary distance, lens maps and entropy of geodesic ows of Finsler metrics, Geom. & Topol. 20 (2016) 469–490]. [ABSTRACT FROM AUTHOR]

We investigate the convexity up to symplectomorphism (called symplectic convexity) of star-shaped toric domains in ℝ 4 . In particular, based on the criterion from Chaidez–Edtmair in [Invent. Math. 229(1) (2022) 243–301] via Ruelle invariant and systolic ratio of the boundary of star-shaped toric domains, we provide elementary operations on domains that can kill the symplectic convexity. These operations only result in small perturbations in terms of domains' volume. Moreover, one of the operations is a systematic way to produce examples of dynamically convex but not symplectically convex toric domains. Finally, we are able to provide concrete bounds for the constants that appear in Chaidez–Edtmair's criterion. [ABSTRACT FROM AUTHOR]

Combining the sharp isoperimetric inequality established by Z. Balogh and A. Kristály [Math. Ann., in press, doi:10.1007/s00208-022-02380-1] with an anisotropic symmetrization argument, we establish sharp Morrey–Sobolev inequalities on n -dimensional Finsler manifolds having nonnegative n -Ricci curvature. A byproduct of this method is a Hardy–Sobolev-type inequality in the same geometric setting. As applications, by using variational arguments, we guarantee the existence/multiplicity of solutions for certain eigenvalue problems and elliptic PDEs involving the Finsler–Laplace operator. Our results are also new in the Riemannian setting. [ABSTRACT FROM AUTHOR]

We obtain curvature estimates for long-time solutions of the continuity method on compact Kähler manifolds with semi-ample canonical line bundles. In this setting, initiated in [G. La Nave and G. Tian, A continuity method to construct canonical metrics, Math. Ann. 365(3) (2016) 911–921; Y. A. Rubinstein, Some discretizations of geometric evolution equations and the Ricci iteration on the space of Kähler metrics, Adv. Math. 218(5) (2008) 1526–1565], we adapt arguments from [F. T.-H. Fong and Y. Zhang, Local curvature estimates of long-time solutions to the Kähler–Ricci flow, Adv. Math. 375 (2020) 107416] for the Kähler–Ricci flow to this setup. As an application, we derive curvature bounds for general metrics on product manifolds. [ABSTRACT FROM AUTHOR]

We solve the Dirichlet problem for complex Monge–Ampère equation near an isolate Klt singularity, which generalizes the result of Eyssidieux et al. [Singular Kähler–Einstein metrics, J. Amer. Math. Soc. 22 (2009) 607–639], where the Monge–Ampère equation is solved on singular varieties without boundary. As a corollary, we construct solutions to Monge–Ampère equation with isolated singularity on strongly pseudoconvex domain Ω contained in ℂ n . [ABSTRACT FROM AUTHOR]

We consider the class of evolution equations of the form u t = F (u , ∂ u / ∂ x , ... , ∂ k u / ∂ x k) , k ≥ 2 , that describe pseudo-spherical surfaces. These were classified by Chern and Tenenblat in [Pseudospherical surfaces and evolution equations, Stud. Appl. Math 74 (1986) 55–83.]. This class of equations is characterized by the property that to each solution of such an equation, there corresponds a 2-dimensional Riemannian metric of constant curvature K = − 1. Motivated by the special properties of the sine-Gordon equation, we investigate the following problem: given such a metric, is there a local isometric immersion in ℝ 3 such that the coefficients of the second fundamental form of the immersed surface depend on a jet of finite order of u ? We extend our earlier results for second-order evolution equations [N. Kahouadji, N. Kamran and K. Tenenblat, Local isometric immersions of pseudo-spherical surfaces and evolution equations, Fields Inst. Commun. 75 (2015) 369–381; N. Kahouadji, N. Kamran and K. Tenenblat, Second-order equations and local isometric immersions of pseudo-spherical surfaces, Comm. Anal. Geom. 24(3) (2016) 605–643.] to k th order equations by proving that there is only one type of equation that admit such an isometric immersion. More precisely, we prove under the condition of finite jet dependency that the coefficients of the second fundamental forms of the local isometric immersion determined by the solutions u are universal, i.e. they are independent of u. Moreover, we show that there exists a foliation of the domain of the parameters of the surface by straight lines with the property that the mean curvature of the surface is constant along the images of these straight lines under the isometric immersion. [ABSTRACT FROM AUTHOR]