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In this paper we are concerned with a two phase boundary obstacle-type problem for the bi-Laplace operator in the upper unit ball. The problem arises in connection with unilateral phenomena for flat elastic plates. It can also be seen as an extension problem to an obstacle problem for the fractional Laplacian $(-\Delta )^{3/2}$, as first observed in \cite{Y}. We establish the well-posedness and the optimal regularity of the solution, and we study the structure of the free boundary. Our proofs are based on monotonicity formulas of Almgren- and Monneau-type., Comment: 26 pages

In this work we consider an inhomogeneous two-phase obstacle-type problem driven by the fractional Laplacian. In particular, making use of the Caffarelli-Silvestre extension, Almgren and Monneau type monotonicity formulas and blow-up analysis, we provide a classification of the possible vanishing orders, which implies the strong unique continuation property. Moreover, we prove a stratification result for the nodal set, together with estimates on its Hausdorff dimensions, for both the regular and the singular part. The main tools come from geometric measure theory and amount to Whitney's Extension Theorem and Federer's Reduction Principle.

The dynamical system described herein uses a hybrid cellular automata (CA) mechanism to attain reversibility, and this approach is adapted to create a novel block cipher algorithm called HCA. CA are widely used for modeling complex systems and employ an inherently parallel model. Therefore, applications derived from CA have a tendency to fit very well in the current computational paradigm where scalability and multi-threading potential are quite desirable characteristics. HCA model has recently received a patent by the Brazilian agency INPI. Several evaluations and analyses performed on the model are presented here, such as theoretical discussions related to its reversibility and an analysis based on graph theory, which reduces HCA security to the well-known Hamiltonian cycle problem that belongs to the NP-complete class. Finally, the cryptographic robustness of HCA is empirically evaluated through several tests, including avalanche property compliance and the NIST randomness suite., Comment: 34 pages, 12 figures

In this paper we give a comprehensive treatment of a two-penalty boundary obstacle problem for a divergence form elliptic operator, motivated by applications to fluid dynamics and thermics. Specifically, we prove existence, uniqueness and optimal regularity of solutions, and establish structural properties of the free boundary. The proofs are based on tailor-made monotonicity formulas of Almgren, Weiss, and Monneau-type, combined with the classical theory of oblique derivative problems.

In this paper we are concerned with a two-penalty boundary obstacle problem of interest in thermics, fluid dynamics and electricity. Specifically, we prove existence, uniqueness and optimal regularity of the solutions, and we establish structural properties of the free boundary., Comment: To appear in Mathematics in Engineering, special issue in honor of Sandro Salsa

In this paper we study the existence, the optimal regularity of solutions, and the regularity of the free boundary near the so-called \emph{regular points} in a thin obstacle problem that arises as the local extension of the obstacle problem for the fractional heat operator $(\partial_t - \Delta_x)^s$ for $s \in (0,1)$. Our regularity estimates are completely local in nature. This aspect is of crucial importance in our forthcoming work on the blowup analysis of the free boundary, including the study of the singular set. Our approach is based on first establishing the boundedness of the time-derivative of the solution. This allows reduction to an elliptic problem at every fixed time level. Using several results from the elliptic theory, including the epiperimetric inequality, we establish the optimal regularity of solutions as well as $H^{1+\gamma,\frac{1+\gamma}{2}}$ regularity of the free boundary near such regular points.

We study the singular set in the thin obstacle problem for degenerate parabolic equations with weight $|y|^a$ for $a \in (-1,1)$. Such problem arises as the local extension of the obstacle problem for the fractional heat operator $(\partial_t - \Delta_x)^s$ for $s \in (0,1)$. Our main result establishes the complete structure and regularity of the singular set of the free boundary. To achieve it, we prove Almgren-Poon, Weiss, and Monneau type monotonicity formulas which generalize those for the case of the heat equation ($a=0$)., Comment: Revised version

In this note, we give a brief overview of obstacle problems for nonlocal operators, focusing on the applications to financial mathematics. The class of nonlocal operators that we consider can be viewed as infinitesimal generators of non-Gaussian asset price models, such as Variance Gamma Processes and Regular L\'evy Processes of Exponential type. In this context, we analyze the existence, uniqueness and regularity of viscosity solutions to obstacle problems which correspond to prices of perpetual and finite expiry American options. Complete proofs can be found in arXiv:1709.10384, where these results have originally appeared., Comment: 12 pages, 1 figure. arXiv admin note: text overlap with arXiv:1709.10384

We prove existence, uniqueness, and regularity of viscosity solutions to the stationary and evolution obstacle problems defined by a class of nonlocal operators that are not stable-like and may have supercritical drift. We give sufficient conditions on the coefficients of the operator to obtain H\"older and Lipschitz continuous solutions. The class of nonlocal operators that we consider include non-Gaussian asset price models widely used in mathematical finance, such as Variance Gamma Processes and Regular L\'evy Processes of Exponential type. In this context, the viscosity solutions that we analyze coincide with the prices of perpetual and finite expiry American options.

We give a comprehensive treatment of the parabolic Signorini problem based on a generalization of Almgren's monotonicity of the frequency. This includes the proof of the optimal regularity of solutions, classification of free boundary points, the regularity of the regular set and the structure of the singular set., Comment: 91 pages, 4 figures

In the recent paper \cite{DGNP} we have proved that the only stable $C^2$ minimal surfaces in the first Heisenberg group $\Hn$ which are graphs over some plane and have empty characteristic locus must be vertical planes. This result represents a sub-Riemannian version of the celebrated theorem of Bernstein. In this paper we extend the result in \cite{DGNP} to $C^2$ complete embedded minimal surfaces in $\mathbb H^1$ with empty characteristic locus. We prove that every such a surface without boundary must be a vertical plane.

We prove a monotonicity result at specific points for the Horizontal Perimeter for a class of surfaces in the Heisenberg group.

We describe the behavior of p-harmonic Green's functions near a singularity in metric measure spaces equipped with a doubling measure and supporting a Poincar\'e inequality.

The purpose of this paper is to establish some basic interior estimates of Cauchy-Schauder type for the heat flow associated with a system of smooth vector fields satisfying Hormander's finite rank condition

We consider various types of Hardy-Sobolev inequalities on a Carnot-Carath\'eodory space $(\Om, d)$ associated to a system of smooth vector fields $X=\{X_1, X_2,...,X_m\}$ on $\RR^n$ satisfying the H\"ormander's finite rank condition $rank Lie[X_1,...,X_m] \equiv n$. One of our main concerns is the trace inequality \int_{\Om}|\phi(x)|^{p}V(x)dx\leq C\int_{\Om}|X\phi|^{p}dx,\qquad \phi\in C^{\infty}_{0}(\Om), where $V$ is a general weight, i.e., a nonnegative locally integrable function on $\Om$, and $1

Let S be a C^2 H-minimal noncharacteristic hypersurface in the first Heisenberg group. We show that if S contains a graphical strip, then it is not a stable minimal surface. Moreover, we show that if S is a C^2 H-minimal noncharacteristic entire graph which is not itself a vertical plane, then S contains a graphical strip. Thus, as a corollary, we obtain an analogue of the Bernstein theorem: the only stable C^2 H-minimal noncharacteristic entire graphs are the vertical planes., Comment: 33 pages

We develope basic geometric quantities and properties of hypersurfaces in Carnot groups.

We provide a partial solution to the isoperimetric problem in the Heisenberg group., Comment: 32 pages, 1 figure

We prove that a family of entire intrinsic minimal graphs in the Heisenberg group are not perimeter minimizing.

Cytoplasm, nucleus and membrane combination techniques for reassembly to form viable amoebae

Existence limits of cells, isoprenes, genetic systems, E. coli, and cell membranes in intracellular relationships

Theoretical cell biology including self-reproductive systems - conferences

Biochemistry, biophysics, cell physiology, and central nervous system research

Statistical mechanics, quantum biochemistry, macromolecule theory, surface and membrane theory, transport phenomena, receptor isolation, and cancer chemotherapy in theoretical biology