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This paper reports the results of the SHREC'14 track: Retrieval and classification on textured 3D models, whose goal is to evaluate the performance of retrieval algorithms when models vary either by geometric shape or texture, or both. The collection to search in is made of 572 textured mesh models, having a two-level classification based on geometry and texture. Together with the dataset, a training set of 96 models was provided. The track saw eight participants and the submission of 22 runs, to either the retrieval or the classification contest, or both. The evaluation results show a promising scenario about textured 3D retrieval methods, and reveal interesting insights in dealing with texture information in the CIELab rather than in the RGB colour space.

We consider local distributional solutions $u\,\epsilon\, W_{loc}^{1,r}\left(\Omega\right)$of non linear elliptic equations of the type \[ -divA\left(x,Du\right)=-div\, f\left(x\right)+g\left(x\right) \] and we prove that $u\,\epsilon\, W_{loc}^{2,r}\left(\Omega\right)$ when r is sufficiently close to 2 which is the exponent related to the growth conditions of the operator (see assumptions (2) and (3)).

In this paper we show approximation procedures for studying singular elliptic problems whose model is $$ \begin{cases} - \Delta u= b(u)|\nabla u|^2+f(x), & \text{in } \; \Omega;\\ u = 0, & \text{on } \; \partial \Omega; \end{cases} $$ where $b(u)$ is singular in the $u$-variable at $u=0$, and $f \in L^m (\Omega)$, with $m>\frac N2$, is a function that does not have a constant sign. We will give an overview of the landscape that occurs when different problems (classified according to the sign of $b(s)$) are considered. So, in each case and using different methods, we will obtain a priori estimates, prove the convergence of the approximate solutions, and show some regularity properties of the limit.

We show optimal existence, nonexistence and regularity results for nonnegative solutions to Dirichlet problems as [Formula: see text] where [Formula: see text] is an open bounded subset of [Formula: see text], [Formula: see text] belongs to [Formula: see text], and [Formula: see text] and [Formula: see text] are continuous functions that may blow up at zero. As a noteworthy fact we show how a non-trivial interaction mechanism between the two nonlinearities [Formula: see text] and [Formula: see text] produces remarkable regularizing effects on the solutions. The sharpness of our main results is discussed through the use of appropriate explicit examples.

In this paper, under very general assumptions, we prove existence and regularity of distributional solutions to homogeneous Dirichlet problems of the form $$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle - \Delta _{1} u = h(u)f &{} \text {in}\, \Omega , \\ u\ge 0&{} \text {in}\ \Omega ,\\ u=0 &{} \text {on}\ \partial \Omega \,. \end{array}\right. } \end{aligned}$$ Here $$\Delta _{1} $$ is the 1-Laplace operator, $$\Omega $$ is a bounded open subset of $$\mathbb {R}^N$$ with Lipschitz boundary, h(s) is a continuous function which may become singular at $$s=0^{+}$$ , and f is a nonnegative datum in $$L^{N,\infty }(\Omega )$$ with suitable small norm. Uniqueness of solutions is also shown provided h is decreasing and $$f>0$$ . As a preparatory tool for our method a general theory for the same problem involving the p-Laplacian (with $$p>1$$ ) as principal part is also established. The main assumptions are further discussed in order to show their optimality.

In this position paper we suggest a possible metric approach to shape comparison that is based on a mathematical formalization of the concept of observer, seen as a collection of suitable operators acting on a metric space of functions. These functions represent the set of data that are accessible to the observer, while the operators describe the way the observer elaborates the data and enclose the invariance that he/she associates with them. We expose this model and illustrate some theoretical reasons that justify its possible use for shape comparison.