Radial symmetry for an elliptic PDE with a free boundary

In this paper we prove symmetry for solutions to the semi-linear elliptic equation Δ u = f ( u ) in B 1 , 0 ≤ u > M , in B 1 , u = M , on ∂ B 1 , \begin{equation*} \Delta u = f(u) \quad \text { in } B_1, \qquad 0 \leq u > M, \quad \text { in } B_1, \qquad u = M, \quad \text { on } \partial B_1, \end{equation*} where M > 0 M>0 is a constant, and B 1 B_1 is the unit ball. Under certain assumptions on the r.h.s. f ( u ) f (u) , the C 1 C^1 -regularity of the free boundary ∂ { u > 0 } \partial \{u>0\} and a second order asymptotic expansion for u u at free boundary points, we derive the spherical symmetry of solutions. A key tool, in addition to the classical moving plane technique, is a boundary Harnack principle (with r.h.s.) that replaces Serrin’s celebrated boundary point lemma, which is not available in our case due to lack of C 2 C^2 -regularity of solutions.

A variant of Clark’s theorem and its applications for nonsmooth functionals without the global symmetric condition

We give a new non-smooth Clark’s theorem without the global symmetric condition. The theorem can be applied to generalized quasi-linear elliptic equations with small continous perturbations. Our results improve the abstract results about a semi-linear elliptic equation in Kajikiya [10] and Li-Liu [11].

Poincaré inequalities and Neumann problems for the variable exponent setting

<abstract><p>In an earlier paper, Cruz-Uribe, Rodney and Rosta proved an equivalence between weighted Poincaré inequalities and the existence of weak solutions to a family of Neumann problems related to a degenerate $ p $-Laplacian. Here we prove a similar equivalence between Poincaré inequalities in variable exponent spaces and solutions to a degenerate $ {p(\cdot)} $-Laplacian, a non-linear elliptic equation with nonstandard growth conditions.</p></abstract>

Mean value theorems for polynomial solutions of linear elliptic equations with constant coefficients in the complex plane

We characterize solutions of the mean value linear elliptic equation with constant coefficients in the complex plane in the case of regular polygon.

On the regularity of very weak solutions for linear elliptic equations in divergence form

In this paper we consider a linear elliptic equation in divergence form $$\begin{aligned} \sum _{i,j}D_j(a_{ij}(x)D_i u )=0 \quad \hbox {in } \Omega . \end{aligned}$$ ∑ i , j D j ( a ij ( x ) D i u ) = 0 in Ω . Assuming the coefficients $$a_{ij}$$ a ij in $$W^{1,n}(\Omega )$$ W 1 , n ( Ω ) with a modulus of continuity satisfying a certain Dini-type continuity condition, we prove that any very weak solution $$u\in L^{n'}_\mathrm{loc}(\Omega )$$ u ∈ L loc n ′ ( Ω ) of (0.1) is actually a weak solution in $$W^{1,2}_\mathrm{loc}(\Omega )$$ W loc 1 , 2 ( Ω ) .

Nonhomogeneous polyharmonic elliptic problems involving GJMS operator on Riemannian manifold

Let [Formula: see text] be a closed Riemannian manifold, under some assumptions on [Formula: see text] and [Formula: see text] by applying a method used in [Tarantello, On nonhomogeneous elliptic equations involving critical Sobolev exponent, Ann. Inst. Henri Poincaré 9 (1992) 281–304], we show the existence and multiplicity of solutions of the semi-linear elliptic equation: [Formula: see text]. When [Formula: see text] is an Einsteinian manifold of positive scalar curvature, under additional conditions, we obtain the existence of positive solutions.