Euler lagrange equations for the following $E(u;f)$

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suppose we have the functional

$$E(u;f) = int_Omega left[ left(u - f right)^2 + g(lVert nabla u rVert^2)lVert nabla u rVert^2right] dxdy = int_Omega mathcalL(x,y,u,u_x,u_y) dxdy$$

Assume all the functions involved are smooth enough, I want to get the euler lagrange equations for the expression above, here $u,f$ are bivariate functions and $g$ is a real function of real variable$

EL equations are given by

$$
fracdEdu = left(fracpartialpartial u - fracd dxfracpartialpartial u_x - fracddyfracpartialpartial u_yright)mathcalL
$$

I'll write this more compactly as
$$
fracdEdu = left(fracpartialpartial u - nabla^T cdot Aright)mathcalL
$$

Where $A$ is defined as the operator

$$
A = beginpmatrix
fracpartialpartial u_x \
fracpartialpartial u_y
endpmatrix =
2 beginpmatrix
u_x \
u_y
endpmatrix fracddlVert nabla urVert^2 = 2 nabla u fracddlVert nabla urVert^2.
$$

Let's observe

$$
fracddlVert nabla urVert^2 mathcalL = fracddlVert nabla urVert^2 left[gleft(lVert nabla u rVert^2 right) lVert nabla u rVert^2 right]
$$

Therefore we can evaluate

$$
nabla^T cdot A mathcalL = 2 nabla^T cdot left( nabla u fracddlVert nabla urVert^2 mathcalL right) = 2nabla^T cdot left[ left(fracddlVert nabla urVert^2 left(gleft(lVert nabla u rVert^2 right) lVert nabla u rVert^2 right) right) nabla uright]
$$

Therefore the Euler-Lgrange equations can be written as

$$
fracdEdu = 2(u - f) - 2nabla^T cdot left[ left(fracddlVert nabla urVert^2 left(gleft(lVert nabla u rVert^2 right) lVert nabla u rVert^2 right) right) nabla uright]
$$

I cannot spot further simplifications, but single quantities should be easily discretized by finite differences.

Is this correct?

integration differential-equations calculus-of-variations euler-lagrange-equation

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edited Aug 1 at 8:45

asked Jul 31 at 10:22

user8469759

1,4271513

add a comment | 

up vote
1
down vote

favorite

suppose we have the functional

$$E(u;f) = int_Omega left[ left(u - f right)^2 + g(lVert nabla u rVert^2)lVert nabla u rVert^2right] dxdy = int_Omega mathcalL(x,y,u,u_x,u_y) dxdy$$

Assume all the functions involved are smooth enough, I want to get the euler lagrange equations for the expression above, here $u,f$ are bivariate functions and $g$ is a real function of real variable$

EL equations are given by

$$
fracdEdu = left(fracpartialpartial u - fracd dxfracpartialpartial u_x - fracddyfracpartialpartial u_yright)mathcalL
$$

I'll write this more compactly as
$$
fracdEdu = left(fracpartialpartial u - nabla^T cdot Aright)mathcalL
$$

Where $A$ is defined as the operator

$$
A = beginpmatrix
fracpartialpartial u_x \
fracpartialpartial u_y
endpmatrix =
2 beginpmatrix
u_x \
u_y
endpmatrix fracddlVert nabla urVert^2 = 2 nabla u fracddlVert nabla urVert^2.
$$

Let's observe

$$
fracddlVert nabla urVert^2 mathcalL = fracddlVert nabla urVert^2 left[gleft(lVert nabla u rVert^2 right) lVert nabla u rVert^2 right]
$$

Therefore we can evaluate

$$
nabla^T cdot A mathcalL = 2 nabla^T cdot left( nabla u fracddlVert nabla urVert^2 mathcalL right) = 2nabla^T cdot left[ left(fracddlVert nabla urVert^2 left(gleft(lVert nabla u rVert^2 right) lVert nabla u rVert^2 right) right) nabla uright]
$$

Therefore the Euler-Lgrange equations can be written as

$$
fracdEdu = 2(u - f) - 2nabla^T cdot left[ left(fracddlVert nabla urVert^2 left(gleft(lVert nabla u rVert^2 right) lVert nabla u rVert^2 right) right) nabla uright]
$$

I cannot spot further simplifications, but single quantities should be easily discretized by finite differences.

Is this correct?

integration differential-equations calculus-of-variations euler-lagrange-equation

share|cite|improve this question

edited Aug 1 at 8:45

asked Jul 31 at 10:22

user8469759

1,4271513

add a comment | 

up vote
1
down vote

favorite

up vote
1
down vote

favorite

suppose we have the functional

$$E(u;f) = int_Omega left[ left(u - f right)^2 + g(lVert nabla u rVert^2)lVert nabla u rVert^2right] dxdy = int_Omega mathcalL(x,y,u,u_x,u_y) dxdy$$

Assume all the functions involved are smooth enough, I want to get the euler lagrange equations for the expression above, here $u,f$ are bivariate functions and $g$ is a real function of real variable$

EL equations are given by

$$
fracdEdu = left(fracpartialpartial u - fracd dxfracpartialpartial u_x - fracddyfracpartialpartial u_yright)mathcalL
$$

I'll write this more compactly as
$$
fracdEdu = left(fracpartialpartial u - nabla^T cdot Aright)mathcalL
$$

Where $A$ is defined as the operator

$$
A = beginpmatrix
fracpartialpartial u_x \
fracpartialpartial u_y
endpmatrix =
2 beginpmatrix
u_x \
u_y
endpmatrix fracddlVert nabla urVert^2 = 2 nabla u fracddlVert nabla urVert^2.
$$

Let's observe

$$
fracddlVert nabla urVert^2 mathcalL = fracddlVert nabla urVert^2 left[gleft(lVert nabla u rVert^2 right) lVert nabla u rVert^2 right]
$$

Therefore we can evaluate

$$
nabla^T cdot A mathcalL = 2 nabla^T cdot left( nabla u fracddlVert nabla urVert^2 mathcalL right) = 2nabla^T cdot left[ left(fracddlVert nabla urVert^2 left(gleft(lVert nabla u rVert^2 right) lVert nabla u rVert^2 right) right) nabla uright]
$$

Therefore the Euler-Lgrange equations can be written as

$$
fracdEdu = 2(u - f) - 2nabla^T cdot left[ left(fracddlVert nabla urVert^2 left(gleft(lVert nabla u rVert^2 right) lVert nabla u rVert^2 right) right) nabla uright]
$$

I cannot spot further simplifications, but single quantities should be easily discretized by finite differences.

Is this correct?

integration differential-equations calculus-of-variations euler-lagrange-equation

share|cite|improve this question

edited Aug 1 at 8:45

asked Jul 31 at 10:22

user8469759

1,4271513

suppose we have the functional

$$E(u;f) = int_Omega left[ left(u - f right)^2 + g(lVert nabla u rVert^2)lVert nabla u rVert^2right] dxdy = int_Omega mathcalL(x,y,u,u_x,u_y) dxdy$$

Assume all the functions involved are smooth enough, I want to get the euler lagrange equations for the expression above, here $u,f$ are bivariate functions and $g$ is a real function of real variable$

EL equations are given by

$$
fracdEdu = left(fracpartialpartial u - fracd dxfracpartialpartial u_x - fracddyfracpartialpartial u_yright)mathcalL
$$

I'll write this more compactly as
$$
fracdEdu = left(fracpartialpartial u - nabla^T cdot Aright)mathcalL
$$

Where $A$ is defined as the operator

$$
A = beginpmatrix
fracpartialpartial u_x \
fracpartialpartial u_y
endpmatrix =
2 beginpmatrix
u_x \
u_y
endpmatrix fracddlVert nabla urVert^2 = 2 nabla u fracddlVert nabla urVert^2.
$$

Let's observe

$$
fracddlVert nabla urVert^2 mathcalL = fracddlVert nabla urVert^2 left[gleft(lVert nabla u rVert^2 right) lVert nabla u rVert^2 right]
$$

Therefore we can evaluate

$$
nabla^T cdot A mathcalL = 2 nabla^T cdot left( nabla u fracddlVert nabla urVert^2 mathcalL right) = 2nabla^T cdot left[ left(fracddlVert nabla urVert^2 left(gleft(lVert nabla u rVert^2 right) lVert nabla u rVert^2 right) right) nabla uright]
$$

Therefore the Euler-Lgrange equations can be written as

$$
fracdEdu = 2(u - f) - 2nabla^T cdot left[ left(fracddlVert nabla urVert^2 left(gleft(lVert nabla u rVert^2 right) lVert nabla u rVert^2 right) right) nabla uright]
$$

I cannot spot further simplifications, but single quantities should be easily discretized by finite differences.

Is this correct?

integration differential-equations calculus-of-variations euler-lagrange-equation

share|cite|improve this question

edited Aug 1 at 8:45

asked Jul 31 at 10:22

user8469759

1,4271513

share|cite|improve this question

share|cite|improve this question

edited Aug 1 at 8:45

edited Aug 1 at 8:45

edited Aug 1 at 8:45

asked Jul 31 at 10:22

user8469759

1,4271513

asked Jul 31 at 10:22

user8469759

1,4271513

asked Jul 31 at 10:22

user8469759

1,4271513

1,4271513

add a comment | 

add a comment | 

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