AN ANISOTROPIC INFINITY LAPLACIAN OBTAINED AS THE LIMIT OF THE ANISOTROPIC (p, q)-LAPLACIAN
AN ANISOTROPIC INFINITY LAPLACIAN OBTAINED AS THE LIMIT OF THE ANISOTROPIC (p, q)-LAPLACIAN
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各向异性无穷大拉普拉斯算子作为各向异性(p,q)-拉普拉斯算子的极限
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发表时间：
2011
期刊：
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通讯作者：
Julio D. Ross
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文献类型：
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作者：
M. Pérez;Julio D. Ross
在这项工作中，我们研究与各向异性 (p, q)-拉普拉斯算子相关的以下狄利克雷问题的解的行为 [ left{egin{array}{ll} -mbox{div}_x(|
abla_x u|^{p-2}
abla_x u) -mbox{div}_y(|
abla_y u|^{q-2}
abla_y u)= 0 & mbox{in}, Omega,\[4pt] u=g & mbox{on}, 部分 Omega,\ end{array}
对了。 ] 为 p, q → ∞。这里 Ω ⊂ ℝN × ℝK 和 $
abla_xu = (frac{partial u}{partial x_1}, frac{partial u}{partial x_2}, 点, frac{partial u}{partial x_N})$ 和 $
abla_yu= (frac{partial u}{partial y_1}, frac{partial u}{partial y_2},dots, frac{partial u}{partial y_K})$表示u相对于前N个变量（x变量）和最后K个变量（y变量）的梯度。我们考虑一个指数序列 (pn, qn)，随着 pn/qn → R 趋于无穷大。我们证明了 un，即 p = pn, q = qn 的解，在 $overline{Omega}$ 中一致地验证了 un → u∞，其中 u∞ 是 [ left{egin{array}{ll} - ...
In this work we study the behavior of the solutions to the following Dirichlet problem related to the anisotropic (p, q)-Laplacian operator [ left{egin{array}{ll} -mbox{div}_x(|
abla_x u|^{p-2}
abla_x u) -mbox{div}_y(|
abla_y u|^{q-2}
abla_y u)= 0 & mbox{in}, Omega,\[4pt] u=g & mbox{on}, partial Omega,\ end{array}
ight. ] as p, q → ∞. Here Ω ⊂ ℝN × ℝK and $
abla_xu = (frac{partial u}{partial x_1}, frac{partial u}{partial x_2}, dots, frac{partial u}{partial x_N})$ and $
abla_yu= (frac{partial u}{partial y_1}, frac{partial u}{partial y_2}, dots, frac{partial u}{partial y_K})$ denote the gradient of u with respect to the first N variables (x variables) and with respect to the last K variables (y variables). We consider a sequence of exponents (pn, qn) that goes to infinity with pn/qn → R. We prove that un, the solution with p = pn, q = qn, verifies un → u∞ uniformly in $overline{Omega}$, where u∞ is the unique viscosity solution to [ left{egin{array}{ll} - ...