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# 数学代写|偏微分方程代考Partial Differential Equations代写|Boundary regularit

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## 数学代写|偏微分方程代考Partial Differential Equations代写|Boundary regularit

To study the regularity of solutions near the boundary, we localize the problem to a neighborhood of a boundary point by use of a partition of unity: We decompose the solution into a sum of functions that are compactly supported in the sets of a suitable open cover of the domain and estimate each function in the sum separately.
Assuming, as in Section 1.10, that the boundary is at least $C^1$, we may ‘flatten’ the boundary in a neighborhood $U$ by a diffeomorphism $\varphi: U \rightarrow V$ that maps $U \cap \Omega$ to an upper half space $V=B_1(0) \cap\left{y_n>0\right}$. If $\varphi^{-1}=\psi$ and $x=\psi(y)$, then by a change of variables (c.f. Theorem 1.44 and Proposition 3.21) the weak formulation (4.34)-(4.35) on $U$ becomes
$$\sum_{i, j=1}^n \int_V \tilde{a}{i j} \frac{\partial \tilde{u}}{\partial y_i} \frac{\partial \tilde{v}}{\partial y_j} d y=\int_V \tilde{f} \tilde{v} d y \quad \text { for all functions } \tilde{v} \in H_0^1(V)$$ where $\tilde{u} \in H^1(V)$. Here, $\tilde{u}=u \circ \psi, \tilde{v}=v \circ \psi$, and $$\tilde{a}{i j}=|\operatorname{det} D \psi| \sum_{p, q=1}^n a_{p q} \circ \psi\left(\frac{\partial \varphi_i}{\partial x_p} \circ \psi\right)\left(\frac{\partial \varphi_j}{\partial x_q} \circ \psi\right), \quad \tilde{f}=|\operatorname{det} D \psi| f \circ \psi .$$
The matrix $\tilde{a}{i j}$ satisfies the uniform ellipticity condition if $a{p q}$ does. To see this, we define $\zeta=\left(D \varphi^t\right) \xi$, or
$$\zeta_p=\sum_{i=1}^n \frac{\partial \varphi_i}{\partial x_p} \xi_i$$
Then, since $D \varphi$ and $D \psi=D \varphi^{-1}$ are invertible and bounded away from zero, we have for some constant $C>0$ that
$$\sum_{i, j=1}^n \tilde{a}{i j} \xi_i \xi_j=|\operatorname{det} D \psi| \sum{p, q=1}^n a_{p q} \zeta_p \zeta_q \geq|\operatorname{det} D \psi| \theta|\zeta|^2 \geq C \theta|\xi|^2$$

## 数学代写|偏微分方程代考Partial Differential Equations代写|Some further perspectives

This book is to a large extent self-contained, with the restriction that the linear theory – Schauder estimates and Campanato theory – is not presented. The reader is expected to be familiar with functional-analytic tools, like the theory of monotone operators. $^4$
The above results give an existence and $L^2$-regularity theory for second-order, uniformly elliptic PDEs in divergence form. This theory is based on the simple a priori energy estimate for $|D u|_{L^2}$ that we obtain by multiplying the equation $L u=f$ by $u$, or some derivative of $u$, and integrating the result by parts.

This theory is a fundamental one, but there is a bewildering variety of approaches to the existence and regularity of solutions of elliptic PDEs. In an attempt to put the above analysis in a broader context, we briefly list some of these approaches and other important results, without any claim to completeness. Many of these topics are discussed further in the references $[\mathbf{9}, \mathbf{1 7}, \mathbf{2 3}]$.
$L^p$-theory: If $1<p<\infty$, there is a similar regularity result that solutions of $L u=f$ satisfy $u \in W^{2, p}$ if $f \in L^p$. The derivation is not as simple when $p \neq 2$, however, and requires the use of more sophisticated tools from real analysis (such as the $L^p$-theory of Calderón-Zygmund operators).
Schauder theory: The Schauder theory provides Hölder-estimates similar to those derived in Section 2.7.2 for Laplace’s equation, and a corresponding existence theory of solutions $u \in C^{2, \alpha}$ of $L u=f$ if $f \in C^{0, \alpha}$ and $L$ has Hölder continuous coefficients. General linear elliptic PDEs are treated by regarding them as perturbations of constant coefficient PDEs, an approach that works because there is no ‘loss of derivatives’ in the estimates of the solution. The Hölder estimates were originally obtained by the use of potential theory, but other ways to obtain them are now known; for example, by the use of Campanato spaces, which provide Hölder norms in terms of suitable integral norms that are easier to estimate directly.
Perron’s method: Perron (1923) showed that solutions of the Dirichlet problem for Laplace’s equation can be obtained as the infimum of superharmonic functions or the supremum of subharmonic functions, together with the use of barrier functions to prove that, under suitable assumptions on the boundary, the solution attains the prescribed boundary values. This method is based on maximum principle estimates.
Boundary integral methods: By the use of Green’s functions, one can often reduce a linear elliptic BVP to an integral equation on the boundary, and then use the theory of integral equations to study the existence and regularity of solutions. These methods also provide efficient numerical schemes because of the lower dimensionality of the boundary.

# 偏微分方程代写

## 数学代写|偏微分方程代考Partial Differential Equations代写|Boundary regularit

$$\sum_{i, j=1}^n \int_V \tilde{a}{i j} \frac{\partial \tilde{u}}{\partial y_i} \frac{\partial \tilde{v}}{\partial y_j} d y=\int_V \tilde{f} \tilde{v} d y \quad \text { for all functions } \tilde{v} \in H_0^1(V)$$哪里$\tilde{u} \in H^1(V)$。这里是$\tilde{u}=u \circ \psi, \tilde{v}=v \circ \psi$和$$\tilde{a}{i j}=|\operatorname{det} D \psi| \sum_{p, q=1}^n a_{p q} \circ \psi\left(\frac{\partial \varphi_i}{\partial x_p} \circ \psi\right)\left(\frac{\partial \varphi_j}{\partial x_q} \circ \psi\right), \quad \tilde{f}=|\operatorname{det} D \psi| f \circ \psi .$$

$$\zeta_p=\sum_{i=1}^n \frac{\partial \varphi_i}{\partial x_p} \xi_i$$

$$\sum_{i, j=1}^n \tilde{a}{i j} \xi_i \xi_j=|\operatorname{det} D \psi| \sum{p, q=1}^n a_{p q} \zeta_p \zeta_q \geq|\operatorname{det} D \psi| \theta|\zeta|^2 \geq C \theta|\xi|^2$$

## 数学代写|偏微分方程代考Partial Differential Equations代写|Some further perspectives

$L^p$ -理论:如果$1<p<\infty$，有一个类似的规律性结果，$L u=f$的解满足$u \in W^{2, p}$如果$f \in L^p$。但是，当$p \neq 2$时推导就不那么简单了，需要使用来自实际分析的更复杂的工具(例如Calderón-Zygmund操作符的$L^p$ -理论)。
Schauder理论:Schauder理论为拉普拉斯方程提供了类似于2.7.2节中推导的Hölder-estimates，如果$f \in C^{0, \alpha}$和$L$具有Hölder连续系数，则提供了对应的$L u=f$解$u \in C^{2, \alpha}$的存在性理论。一般的线性椭圆偏微分方程被视为常系数偏微分方程的扰动，这种方法是有效的，因为在解的估计中没有“导数损失”。Hölder估计值最初是通过使用势理论获得的，但现在已知其他获得它们的方法;例如，通过使用Campanato空间，它提供了Hölder范数，用合适的积分范数表示，更容易直接估计。
Perron方法:Perron(1923)证明了拉普拉斯方程的Dirichlet问题的解可以用超调和函数的极值或次调和函数的极值来求得，并利用势垒函数证明了在边界上适当的假设下，解达到规定的边界值。该方法基于最大原理估计。

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