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Register a new userOn stiff problems via Dirichlet forms
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The stiff problem is concerned with a thermal conduction model with a singular barrier of zero volume. In this paper, we shall build the phase transitions for the stiff problems in one-dimensional space. It turns out that every phase transition definitely depends on the total thermal resistance of the barrier, and the three phases correspond to the so-called impermeable pattern, semi-permeable pattern and permeable pattern of thermal conduction respectively. For each pattern, the related boundary condition of the flux at the barrier is also derived. Mathematically, we shall introduce and explore the so-called snapping out Markov process, which is the probabilistic counterpart of semi-permeable pattern in the stiff problem.
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This paper establishes explicit solutions for fractional diffusion problems on bounded domains. It also gives stochastic solutions, in terms of Markov processes time-changed by an inverse stable subordinator whose index equals the order of the fractional time derivative. Some applications are given, to demonstrate how to specify a well-posed Dirichlet problem for space-time fractional diffusions in one or several variables. This solves an open problem in numerical analysis.
In the context of a metric measure Dirichlet space satisfying volume doubling and Poincare inequality, we prove the parabolic Harnack inequality for weak solutions of the heat equation associated with local nonsymmetric bilinear forms. In particular, we show that these weak solutions are locally bounded.
We construct symmetric self-similar Dirichlet forms on unconstrained Sierpinski carpets, which are natural extension of planner Sierpinski carpets by allowing the small cells to live off the $1/k$ grids. The intersection of two cells can be a line segment of irrational length, and we also drop the non-diagonal assumption in this recurrent setting. The proof of the existence is purely analytic. A uniqueness theorem is also provided. Moreover, the additional freedom of unconstrained Sierpinski carpets allows us to slide the cells around. In this situation, we view unconstrained Sierpinski carpets as moving fractals, and we prove that the self-similar Dirichlet forms will vary continuously in a $Gamma$-convergence sense.
We study harmonic functions for general Dirichlet forms. First we review consequences of Fukushimas ergodic theorem for the harmonic functions in the domain of the $ L^{p} $ generator. Secondly we prove analogues of Yaus and Karps Liouville theorems for weakly harmonic functions. Both say that weakly harmonic functions which satisfy certain $ L^{p} $ growth criteria must be constant. As consequence we give an integral criterion for recurrence.
In this paper we will study a stiff problem in two-dimensional space and especially its probabilistic counterpart. Roughly speaking, the heat equation with a parameter $varepsilon>0$ is under consideration: [ partial_t u^varepsilon(t,x)=frac{1}{2} abla cdot left(mathbf{A}_varepsilon(x) abla u^varepsilon(t,x) right),quad tgeq 0, xin mathbb{R}^2, ] where $mathbf{A}_varepsilon(x)=text{Id}_2$, the identity matrix, for $x otin Omega_varepsilon:={x=(x_1,x_2)in mathbb{R}^2: |x_2|<varepsilon}$ while $$mathbf{A}_varepsilon(x):=begin{pmatrix} a_varepsilon^- & 0 0 & a^shortmid_varepsilon end{pmatrix}$$ with two positive constants $a^-_varepsilon, a^shortmid_varepsilon$ for $xin Omega_varepsilon$. There exists a diffusion process $X^varepsilon$ on $mathbb{R}^2$ associated to this heat equation in the sense that $u^varepsilon(t,x):=mathbf{E}^xu^varepsilon(0,X_t^varepsilon)$ is its unique weak solution. Note that $Omega_varepsilon$ collapses to the $x_1$-axis, a barrier of zero volume, as $varepsilondownarrow 0$. The main purpose of this paper is to derive all possible limiting process $X$ of $X^varepsilon$ as $varepsilondownarrow 0$. In addition, the limiting flux $u$ of the solution $u^varepsilon$ as $varepsilondownarrow 0 $ and all possible boundary conditions satisfied by $u$ will be also characterized.
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