Elliptic partial differential equation

Elliptic partial differential equations are a special class of partial differential equations (PDG). They are formulated with the help of elliptic differential operators . The solutions of an elliptical partial differential equation have certain properties, which are explained in more detail here. The Laplace operator is probably the best known elliptic differential operator , and Poisson's equation is the corresponding partial differential equation. ${\displaystyle Pu=f}$

Physical interpretation

The elliptic differential equation is a generalization of Laplace's equation and Poisson's equation. A second order elliptic differential equation has the form

${\displaystyle P(u)(x)=\sum _{i,j}^{n}a_{ij}(x)\partial _{x_{i},x_{j}}^{2}u(x)+\sum _{i}^{n}b_{i}(x)\partial _{x_{i}}u(x)+c(x)u(x)=f(x)}$,

where the coefficient functions , and must satisfy suitable conditions. ${\displaystyle a_{ij}}$ ${\displaystyle b_{i}}$ ${\displaystyle c}$

Such differential equations typically occur in connection with stationary (time-independent) problems. They often describe a state of minimal energy. The mentioned Laplace and Poisson equations describe the temperature distribution in a body or the electrostatic charge distribution in a body. Other elliptical differential equations are used, for example, to study the concentration of certain chemical substances. The terms of order two describe the diffusion . The first-order terms describe the transport, and the zero-order term describes the local increase and decrease.

Non-linear elliptic differential equations also occur in the calculus of variations and differential geometry .

definition

Elliptic differential operator

A differential operator , listed in Multi index notation , the order in an area called the point elliptical, if for all true ${\displaystyle \textstyle P(u)(x)=\sum _{|\alpha |\leq m}a_{\alpha }(x)\partial _{x}^{\alpha }u(x)}$ ${\displaystyle m}$ ${\displaystyle \Omega \subset \mathbb {R} ^{n}}$ ${\displaystyle y\in \Omega }$ ${\displaystyle \xi \in \mathbb {R} ^{n}\backslash \{0\}}$

${\displaystyle P_{m}(y,\xi ):=\sum _{|\alpha |=m}a_{\alpha }(y)\xi ^{\alpha }\neq 0\ .}$

One calls the main symbol of . A differential operator is called elliptic if it is elliptic for all . ${\displaystyle \ P_{m}}$ ${\displaystyle P}$ ${\displaystyle y\in \Omega }$

Elliptic differential equation

Let be an elliptic differential operator and a function, then the equation is called ${\displaystyle P}$ ${\displaystyle f}$

${\displaystyle Pu=f}$

elliptical differential equation and is the function we are looking for in this differential equation. ${\displaystyle u}$

Uniformly elliptic differential operator

A differential operator is called uniformly elliptic in if there is a such that ${\displaystyle P}$ ${\displaystyle U}$ ${\displaystyle c>0}$

${\displaystyle |P_{m}(y,\xi )|\geq c|\xi |^{m}}$

applies to all . ${\displaystyle (y,\xi )\in U\times \mathbb {R} ^{n}}$

Hypo-elliptic differential operator

An operator with constant coefficients is called hypo-elliptic if there is one such that for all with and all : ${\displaystyle \textstyle P(D)=\sum _{|\alpha |\leq m}a_{\alpha }D^{\alpha }}$ ${\displaystyle a_{\alpha }\in \mathbb {C} }$ ${\displaystyle C>0}$ ${\displaystyle \xi \in \mathbb {R} ^{n}}$ ${\displaystyle |\xi |\geq C}$ ${\displaystyle \alpha \in \mathbb {N} ^{n}}$

• ${\displaystyle P(\xi )\neq 0}$ and
• ${\displaystyle |D^{\alpha }P(\xi )|\leq C|P(\xi )|\cdot |\xi |^{-|\alpha |}}$.

More generally, a differential operator on an open set with not necessarily constant coefficients is called hypo-elliptic , if open for every set , bounded and every distribution its implication ${\displaystyle P(D)}$ ${\displaystyle U\subset \mathbb {R} ^{d}}$ ${\displaystyle U'\subset U}$ ${\displaystyle u\in {\mathcal {D}}(U')}$

${\displaystyle Pu\in C^{\infty }(U')\quad \Rightarrow \quad u\in C^{\infty }(U')}$

applies. In words: If the image is infinitely differentiable in the distribution sense of the differential operator , this already applies to the archetypes. ${\displaystyle P}$

In contrast to the uniformly elliptic differential operator, the hypo-elliptic differential operator is a generalization of the elliptic differential operator. This requirement of the differential operator is therefore weaker. See the regularity theory of elliptic operators below.

Origin of name

The adjective elliptic in the name elliptic partial differential equation comes from the theory of conic sections . In this theory, in the case, the solution set , the equation ${\displaystyle B^{2}-4AC<0}$

${\displaystyle Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0}$

Called an ellipse . If we now consider the homogeneous differential equation

${\displaystyle A\partial _{x_{1},x_{1}}^{2}u(x)+B\partial _{x_{1},x_{2}}^{2}u(x)+C\partial _{x_{2},x_{2}}^{2}u(x)+D\partial _{x_{1}}u(x)+E\partial _{x_{2}}u(x)+Fu(x)=0}$

second order in two dimensions with constant coefficients, then this is uniformly elliptical if and only if applies. ${\displaystyle B^{2}-4AC<0}$

Examples

• The most important example of a uniformly elliptic differential operator is the Laplace operator

${\displaystyle \Delta u(x)=\sum _{j=1}^{n}\partial _{x_{j}x_{j}}^{2}u(x),}$

whose main symbol is. Functions that satisfy the Laplace equation are called harmonic and have some special properties, such as the fact that they can be differentiated any number of times. One now has the hope that these properties can be transferred to “similar” differential operators. ${\displaystyle P_{2}(y,\xi )=((-1)\cdot \xi _{1}^{2}+\ldots +(-1)\cdot \xi _{n}^{2})=-|\xi |^{2}}$ ${\displaystyle \Delta u=0}$

• The Cauchy-Riemann operator

${\displaystyle {\frac {\partial }{\partial {\bar {z}}}}={\frac {1}{2}}\left({\frac {\partial }{\partial x}}+i{\frac {\partial }{\partial y}}\right)}$

is uniformly elliptical because its main symbol is . ${\displaystyle {\tfrac {1}{2}}\left(i\xi _{1}-\xi _{2}\right)}$

• The parabolic partial differential operator is hypo-elliptical, but not uniformly elliptical. The parabolic differential equation is called the heat conduction equation . ${\displaystyle P=D_{t}-\Delta _{x}}$ ${\displaystyle Pu=0}$

Theory of elliptic differential equations of the second order

In the following, the most important statements for elliptic differential operators of order two in dimensions are shown. So be ${\displaystyle n}$

${\displaystyle P(u)(x)=\sum _{i,j}^{n}a_{ij}(x)\partial _{x_{i},x_{j}}^{2}u(x)+\sum _{i}^{n}b_{i}(x)\partial _{x_{i}}u(x)+c(x)u(x)}$

an elliptic differential operator of order two. In addition, let it be an open , connected , bounded subset with a Lipschitz boundary . ${\displaystyle U\subset \mathbb {R} ^{n}}$

Existence statement

The coefficient functions are all measurable and limited functions. Then for each there is a unique weak solution of the Dirichlet boundary value problem ${\displaystyle a_{ij},b_{i},c}$ ${\displaystyle f\in L^{2}(U)}$ ${\displaystyle u\in H_{0}^{1}(U)}$

${\displaystyle \left\{{\begin{array}{cc}Pu=f&{\text{in}}\ U\\u=0&{\text{in}}\ \partial U,\end{array}}\right.}$

if the bilinear form associated with the differential operator is coercive . Here is defined capacity ${\displaystyle P}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle {\mathcal {P}}:H_{0}^{1}(U)\times H_{0}^{1}(U)\rightarrow \mathbb {R} }$

${\displaystyle {\mathcal {P}}(u,\varphi ):=\int _{U}\left(-\sum _{i,j}^{n}a_{ij}(x)\partial _{x_{i}}u(x)\partial _{x_{j}}\varphi (x)\right)+\left(\sum _{i}^{n}b_{i}(x)\partial _{x_{i}}u(x)\varphi (x)\right)+c(x)u(x)\varphi (x){\rm {d}}x}$.

With the Lax-Milgram lemma one deduces the existence and the uniqueness of the solution from the bilinear form . If uniformly elliptical, the associated bilinear form is always coercive. If a Neumann boundary condition is used instead of a Dirichlet boundary condition , then, if the associated bilinear form is again coercive, there is exactly one solution of the partial differential equation, which can be proven almost in the same way. ${\displaystyle u}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle P}$ ${\displaystyle {\mathcal {P}}}$

Regularity

Be for all , and also be and a weak solution of the elliptic differential equation ${\displaystyle a_{ij},b_{i},c\in C^{\infty }(U)}$ ${\displaystyle i,j}$ ${\displaystyle f\in C^{\infty }(U)}$ ${\displaystyle u\in H^{1}(U)}$

${\displaystyle Pu=f\ {\text{in}}\ U}$.

Then applies . ${\displaystyle u\in C^{\infty }(U)}$

Maximum principle

A maximum principle applies to elliptic differential operators of the second order . Be in and be . ${\displaystyle c\geq 0}$ ${\displaystyle U}$ ${\displaystyle u\in C^{2}(U)\cap C({\overline {U}})}$

1. If

${\displaystyle Pu\leq 0\ {\text{in}}\ U}$

holds and assumes a nonnegative maximum in an inner point of , then is constant. ${\displaystyle u}$ ${\displaystyle {\overline {U}}}$ ${\displaystyle u}$

2. If

${\displaystyle Pu\geq 0\ {\text{in}}\ U}$

holds and assumes a non-positive minimum in an inner point of , then is constant. ${\displaystyle u}$ ${\displaystyle {\overline {U}}}$ ${\displaystyle u}$

Eigenvalue problems

Consider the boundary value problem

${\displaystyle Pu=\lambda u\ {\text{in}}\ U}$

${\displaystyle \ u=0\ {\text{in}}\ \partial U,}$

where is an eigenvalue of the differential operator . In addition, let be a symmetric differential operator. ${\displaystyle \lambda }$ ${\displaystyle P}$ ${\displaystyle P}$

1. Then all eigenvalues ​​are real. ${\displaystyle P}$

2. In addition, all eigenvalues ​​have the same sign and only have a finite multiplicity.

3. Finally there is an orthonormal basis of with as an eigenfunction of the eigenvalue . ${\displaystyle \{w_{k}\}_{k=1}^{\infty }}$ ${\displaystyle L^{2}(U)}$ ${\displaystyle w_{k}\in H_{0}^{1}(U)}$ ${\displaystyle \lambda _{k}}$

Theory of elliptic pseudodifferential operators

definition

A pseudo differential operator is called elliptic if its symbol is actually carried and the homogeneous main symbol is uniformly elliptic - or equivalent if the inequality for a constant for and holds in a conical neighborhood of for the real symbol . ${\displaystyle p\in S_{cl}^{m}(X)}$ ${\displaystyle V}$ ${\displaystyle (x_{0},\xi _{0})}$ ${\displaystyle |a(x,\xi )|\geq {\tfrac {1}{C}}(1+|\xi |)^{m}}$ ${\displaystyle C>0}$ ${\displaystyle (x,\xi )\in V}$ ${\displaystyle |\xi |\geq C}$

Invertibility

Let be an elliptical pseudodifferential operator and , then there exists a actually carried pseudodifferential operator such that ${\displaystyle P\in \Psi _{\rho ,\delta }^{m}(X)}$ ${\displaystyle \rho >\delta }$ ${\displaystyle Q\in \Psi _{\rho ,\delta }^{-m}(X)}$

${\displaystyle (P\circ Q)(u)=(Q\circ P)(u)=I(u)+R(u)}$

applies. The identity operator is , and is an operator that maps every distribution to a smooth function. This operator is called Parametrix . The operator can therefore be inverted modulo . This property makes the elliptic pseudo differential operator and thus, as a special case, the elliptic differential operator, a Fredholm operator . ${\displaystyle I}$ ${\displaystyle R\in \Psi _{\rho ,\delta }^{-\infty }(X)}$ ${\displaystyle Q}$ ${\displaystyle P}$ ${\displaystyle \Psi _{\rho ,\delta }^{-\infty }(X)}$

Singular carrier

Let again be an elliptic pseudo differential operator and . Then applies to every distribution ${\displaystyle P\in \Psi _{\rho ,\delta }^{m}(X)}$ ${\displaystyle \rho >\delta }$ ${\displaystyle u\in {\mathcal {D}}'(X)}$

${\displaystyle \operatorname {sing\ supp} (Pu)=\operatorname {sing\ supp} (u).}$

The singular carrier of a distribution does not change.

literature

• Gerhard Dziuk: Theory and Numerics of Partial Differential Equations , de Gruyter, Berlin 2010, ISBN 978-3-11-014843-5 , pages 151-181.
• Lawrence Craig Evans : Partial Differential Equations , American Mathematical Society, Providence 2002, ISBN 0-8218-0772-2 .
• Alain Grigis & Johannes Sjöstrand - Microlocal Analysis for Differential Operators , Cambridge University Press, 1994, ISBN 0-521-44986-3 .

Individual evidence

1. ↑ Alain Grigis, Johannes Sjöstrand - Micro Local Analysis for Differential Operators . Cambridge University Press, 1994, ISBN 0-521-44986-3 , p. 41.