Let $L$ be an elliptic linear operator in a domain in ${\mathbf{R}}^n$. We imposse only weak regularity conditions on the coefficients. Then the adjoint $L^{*}$ exists in the sense of distributions, and we start by deducing a regularity theorem for distribution solutions of equations of type $L^{*}u=$ given distribution. We then apply to $L^{*}$ R.M. Hervé’s theory of adjoint harmonic spaces. Some other properties of $L^{*}$ are also studied. The results generalize earlier work of the author.

We consider pure mth order subcoercive operators with complex coefficients acting on a connected nilpotent Lie group. We derive Gaussian bounds with the correct small time singularity and the optimal large time asymptotic behaviour on the heat kernel and all its derivatives, both right and left. Further we prove that the Riesz transforms of all orders are bounded on the Lp -spaces with p ∈ (1, ∞). Finally, for second-order operators with real coefficients we derive matching Gaussian lower bounds...

Il est bien connu qu’une fonction $f$ sur ${ℝ}^n$ est harmonique - Δf = 0 - si et seulement si sa moyenne sur toute sphère est égale à sa valeur au centre de cette sphère. De manière semblable, f vérifie l’équation de Helmholtz Δf + cf = 0 si et seulement si sa moyenne sur la sphère de centre x et de rayon r vaut $\Gamma (n/2){(r\surd c/2)}^{(2-n)/2}{J}_{(n-2)/2}\left(r\surd c\right)·f\left(x\right)$. Dans ce travail, nous généralisons ces résultats à l’opérateur ${(\Delta +c)}^k$ où k est un entier strictement positif et c une constante non nulle. Bien qu’une méthode pour y parvenir soit esquissée dans...

Using a version of the Local Linking Theorem and the Fountain Theorem, we obtain some existence and multiplicity results for a class of superquadratic elliptic equations.

In this work, by using the Mountain Pass Theorem, we give a result on the existence of solutions concerning a class of nonlocal $p$-Laplacian Dirichlet problems with a critical nonlinearity and small perturbation.

In this paper we establish a Liouville type theorem for fully nonlinear elliptic equations related to a conjecture of De Giorgi in ${ℝ}^2$. We prove that if the level lines of a solution have bounded curvature, then these level lines are straight lines. As a consequence, the solution is one-dimensional. The method also provides a result on free boundary problems of Serrin type.

Qualitative comparison of the nonoscillatory behavior of the equations $$L_{n}y\left(t\right)+H(t,y\left(t\right))=0$$ and $$L_{n}y\left(t\right)+H(t,y\left(g\left(t\right)\right))=0$$ is sought by way of finding different nonoscillation criteria for the above equations. $L_n$ is a disconjugate operator of the form $$L_n=\frac{1}{p_n\left(t\right)}\frac{\mathrm{d}}{\mathrm{d}t}\frac{1}{p_{n-1}\left(t\right)}\frac{\mathrm{d}}{\mathrm{d}t}...\frac{\mathrm{d}}{\mathrm{d}t}\frac{·}{p_0\left(t\right)}.$$ Both canonical and noncanonical forms of $L_n$ have been studied.