For an open set V ⊂Cⁿ, denote by Mα(V) the family of α-analytic functions that obey a boundary maximum modulus principle. We prove that, on a bounded “harmonically fat” domain Ω ⊂ Cⁿ, a function f ∈ M_α(Ω\ f⁻¹(0)) automatically satisfies f ∈ M_α(Ω), if it is C^αj−1-smooth in the zj variable, α ∈ Zⁿ₊ up to the boundary. For a submanifold U⊂Cⁿ, denote by M_α(U), the set of functions locally approximable by α-analytic functions where each approximating member and its reciprocal (off the singularities) obey the boundary maximum modulus principle. We prove, that for a C³-smooth hypersurface, Ω, a member of M_α(Ω), cannot have constant modulus near a point where the Levi form has a positive eigenvalue, unless it is there the trace of a polyanalytic function of a simple form. The result can be partially generalized to C⁴-smooth submanifolds of higher codimension, at least near points with a Levi cone condition.

In this paper we prove the existence of an infinite number of radial solutions of $\Delta u + K(r)f(u)= 0$ on the exterior of the ball of radius $R>0$ centered at the origin in ${\mathbb R}^{N}$ where $f$ is odd with $f<0$ on $(0, \beta) $, $f>0$ on $(\beta, \delta),$  $f\equiv 0$ for $u> \delta$, and where the function $K(r)$ is assumed to be positive and $K(r)\to 0$ as $r \to \infty.$ The primitive $F(u)  = \int_{0}^{u} f(s) \, {\rm d}s$ has a  ``hilltop'' at $u=\delta$ which allows one to use the shooting method to prove the existence of solutions.

Given f ,g two C^r,α either symplectic forms or volume forms on a bounded open set Ω ⊂Rⁿ with 0<α<1 and r ≥0, we give natural conditions for the existence of a map ϕ∈Diff^r+1,α(Ω;Ω) satisfying $$ϕ ∗ (g) = f \;\; {\rm in}  \;\; Ω \quad  {\rm\ and \quad supp}  \;\;(ϕ−id)⊂Ω.$$

In this paper, we discuss the local and global existence of weak solutions for some hyperbolic-parabolic systems modeling chemotaxis in R^N.

In this paper, we investigate the Cahn-Hilliard equation defined on the half space and subject to the Neumann boundary and initial condition. For given initial data in some sobolev space, we prove the existence and analytic smoothing effect of the solution.