ترغب بنشر مسار تعليمي؟ اضغط هنا

On the Perron solution of the caloric Dirichlet problem: an elementary approach

133   0   0.0 ( 0 )
 نشر من قبل Alessia Elisabetta Kogoj
 تاريخ النشر 2021
  مجال البحث
والبحث باللغة English




اسأل ChatGPT حول البحث

By an easy trick taken from caloric polynomial theory we construct a family $mathscr{B}$ of $almost regular$ domains for the caloric Dirichlet problem. $mathscr{B}$ is a basis of the Euclidean topology. This allows to build, with a basically elementary procedure, the Perron solution to the caloric Dirichlet problem on every bounded domain.



قيم البحث

اقرأ أيضاً

101 - Alessia E. Kogoj 2016
We show how to apply harmonic spaces potential theory in the study of the Dirichlet problem for a general class of evolution hypoelliptic partial differential equations of second order. We construct Perron-Wiener solution and we provide a sufficient condition for the regularity of the boundary points. Our criterion extends and generalizes the classical parabolic-cone criterion for the Heat equation due to Effros and Kazdan.
This paper investigates sufficient conditions for a Feynman-Kac functional up to an exit time to be the generalized viscosity solution of a Dirichlet problem. The key ingredient is to find out the continuity of exit operator under Skorokhod topology, which reveals the intrinsic connection between overfitting Dirichlet boundary and fine topology. As an application, we establish the sub and supersolutions for a class of non-stationary HJB (Hamilton-Jacobi-Bellman) equations with fractional Laplacian operator via Feynman-Kac functionals associated to $alpha$-stable processes, which help verify the solvability of the original HJB equation.
We shall discuss the inhomogeneous Dirichlet problem for: $f(x,u, Du, D^2u) = psi(x)$ where $f$ is a natural differential operator, with a restricted domain $F$, on a manifold $X$. By natural we mean operators that arise intrinsically from a given ge ometry on $X$. An important point is that the equation need not be convex and can be highly degenerate. Furthermore, the inhomogeneous term can take values at the boundary of the restricted domain $F$ of the operator $f$. A simple example is the real Monge-Amp`ere operator ${rm det}({rm Hess},u) = psi(x)$ on a riemannian manifold $X$, where ${rm Hess}$ is the riemannian Hessian, the restricted domain is $F = {{rm Hess} geq 0}$, and $psi$ is continuous with $psigeq0$. A main new tool is the idea of local jet-equivalence, which gives rise to local weak comparison, and then to comparison under a natural and necessary global assumption. The main theorem applies to pairs $(F,f)$, which are locally jet-equivalent to a given constant coefficient pair $({bf F}, {bf f})$. This covers a large family of geometric equations on manifolds: orthogonally invariant operators on a riemannian manifold, G-invariant operators on manifolds with G-structure, operators on almost complex manifolds, and operators, such as the Lagrangian Monge-Amp`ere operator, on symplectic manifolds. It also applies to all branches of these operators. Complete existence and uniqueness results are established with existence requiring the same boundary assumptions as in the homogeneous case [10]. We also have results where the inhomogeneous term $psi$ is a delta function.
We solve the nonlinear Dirichlet problem (uniquely) for functions with prescribed asymptotic singularities at a finite number of points, and with arbitrary continuous boundary data, on a domain in euclidean space. The main results apply, in particula r, to subequations with a Riesz characteristic $p geq 2$. In this case it is shown that, without requiring uniform ellipticity, the Dirichlet problem can be solved uniquely for arbitrary continuous boundary data with singularities asymptotic to the Riesz kernel: $Theta_j K_p(x - x_j)$, where $K_p(x) = - {1over|x|^{p-2}}$ for $p>2$ and $K_2(x) = log |x|$, at any prescribed finite set of points $x_1,...,x_k$ in the domain and any finite set of positive real numbers $Theta_1,..., Theta_k$. This sharpens a previous result of the authors concerning the discreteness of high-density sets of subsolutions. Uniqueness and existence results are also established for finite-type singularities such as $Theta_j |x - x_j|^{2-p}$ for $1leq p<2$. The main results apply similarly with prescribed singularities asymptotic to the fundamental solutions of Armstrong-Sirakov-Smart (in the uniformly elliptic case).
In this paper, we solve the Dirichlet problem with continuous boundary data for the Lagrangian mean curvature equation on a uniformly convex, bounded domain in $mathbb{R}^n$.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا