How to treat an equation of the form $-Delta u=Gcdot nabla u+f(u) ?$
$begingroup$
There are plenty of variational techniques (direct methods of calculus of variations, mountain pass type theorems, Lusternik-Schnirelmann theory) to prove the existence of solutions of a semilinear elliptic equation of the form $$-Delta u=f(u)$$
in the Sobolev space $H^1_0(Omega)$, where $Omega$ is a non-empty open subset of $mathbb{R}^n$,
under suitable hypothesis on $f:mathbb{R}tomathbb{R}$, thanks to the fact that we can see weak solutions of this problem as the stationary points of the functional:
$$I:H^1_0(Omega)tomathbb{R}, umapstofrac{1}{2}|u|_{H^1_0}^2-int_Omegaint_0^{u(x)}f(s)operatorname{d}soperatorname{d}x.$$
If $G:Omegarightarrowmathbb{R}^n$, how can we treat the equation:
$$-Delta u=Gcdotnabla u+f(u),$$
or, more generally, if $g:mathbb{R}^ntomathbb{R}$, the equation:
$$-Delta u=gleft(nabla uright)+f(u)?$$
If $n=1$, I saw in the $G$-case that we can transform the equation into another semilinear elliptic equation that hasn't the dissipative term $u'$, with the same trick used in Sturm-Liouville theory, and so we can bring back this problem into the realm of the previous variational problem. However, what about the $g$-case if $n=1$? What about the $G$-case if $nge2$? Can we say anything about the $g$-case if $nge 2$?
pde calculus-of-variations variational-analysis
asked Dec 27 '18 at 8:28
BobBob
1,6151725
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add a comment |
$begingroup$
There are plenty of variational techniques (direct methods of calculus of variations, mountain pass type theorems, Lusternik-Schnirelmann theory) to prove the existence of solutions of a semilinear elliptic equation of the form $$-Delta u=f(u)$$
in the Sobolev space $H^1_0(Omega)$, where $Omega$ is a non-empty open subset of $mathbb{R}^n$,
under suitable hypothesis on $f:mathbb{R}tomathbb{R}$, thanks to the fact that we can see weak solutions of this problem as the stationary points of the functional:
$$I:H^1_0(Omega)tomathbb{R}, umapstofrac{1}{2}|u|_{H^1_0}^2-int_Omegaint_0^{u(x)}f(s)operatorname{d}soperatorname{d}x.$$
If $G:Omegarightarrowmathbb{R}^n$, how can we treat the equation:
$$-Delta u=Gcdotnabla u+f(u),$$
or, more generally, if $g:mathbb{R}^ntomathbb{R}$, the equation:
$$-Delta u=gleft(nabla uright)+f(u)?$$
If $n=1$, I saw in the $G$-case that we can transform the equation into another semilinear elliptic equation that hasn't the dissipative term $u'$, with the same trick used in Sturm-Liouville theory, and so we can bring back this problem into the realm of the previous variational problem. However, what about the $g$-case if $n=1$? What about the $G$-case if $nge2$? Can we say anything about the $g$-case if $nge 2$?
pde calculus-of-variations variational-analysis
asked Dec 27 '18 at 8:28
BobBob
1,6151725
$endgroup$
add a comment |
$begingroup$
There are plenty of variational techniques (direct methods of calculus of variations, mountain pass type theorems, Lusternik-Schnirelmann theory) to prove the existence of solutions of a semilinear elliptic equation of the form $$-Delta u=f(u)$$
in the Sobolev space $H^1_0(Omega)$, where $Omega$ is a non-empty open subset of $mathbb{R}^n$,
under suitable hypothesis on $f:mathbb{R}tomathbb{R}$, thanks to the fact that we can see weak solutions of this problem as the stationary points of the functional:
$$I:H^1_0(Omega)tomathbb{R}, umapstofrac{1}{2}|u|_{H^1_0}^2-int_Omegaint_0^{u(x)}f(s)operatorname{d}soperatorname{d}x.$$
If $G:Omegarightarrowmathbb{R}^n$, how can we treat the equation:
$$-Delta u=Gcdotnabla u+f(u),$$
or, more generally, if $g:mathbb{R}^ntomathbb{R}$, the equation:
$$-Delta u=gleft(nabla uright)+f(u)?$$
If $n=1$, I saw in the $G$-case that we can transform the equation into another semilinear elliptic equation that hasn't the dissipative term $u'$, with the same trick used in Sturm-Liouville theory, and so we can bring back this problem into the realm of the previous variational problem. However, what about the $g$-case if $n=1$? What about the $G$-case if $nge2$? Can we say anything about the $g$-case if $nge 2$?
pde calculus-of-variations variational-analysis
asked Dec 27 '18 at 8:28
BobBob
1,6151725
$endgroup$
There are plenty of variational techniques (direct methods of calculus of variations, mountain pass type theorems, Lusternik-Schnirelmann theory) to prove the existence of solutions of a semilinear elliptic equation of the form $$-Delta u=f(u)$$
in the Sobolev space $H^1_0(Omega)$, where $Omega$ is a non-empty open subset of $mathbb{R}^n$,
under suitable hypothesis on $f:mathbb{R}tomathbb{R}$, thanks to the fact that we can see weak solutions of this problem as the stationary points of the functional:
$$I:H^1_0(Omega)tomathbb{R}, umapstofrac{1}{2}|u|_{H^1_0}^2-int_Omegaint_0^{u(x)}f(s)operatorname{d}soperatorname{d}x.$$
If $G:Omegarightarrowmathbb{R}^n$, how can we treat the equation:
$$-Delta u=Gcdotnabla u+f(u),$$
or, more generally, if $g:mathbb{R}^ntomathbb{R}$, the equation:
$$-Delta u=gleft(nabla uright)+f(u)?$$
If $n=1$, I saw in the $G$-case that we can transform the equation into another semilinear elliptic equation that hasn't the dissipative term $u'$, with the same trick used in Sturm-Liouville theory, and so we can bring back this problem into the realm of the previous variational problem. However, what about the $g$-case if $n=1$? What about the $G$-case if $nge2$? Can we say anything about the $g$-case if $nge 2$?
pde calculus-of-variations variational-analysis
pde calculus-of-variations variational-analysis
asked Dec 27 '18 at 8:28
BobBob
1,6151725
asked Dec 27 '18 at 8:28
BobBob
1,6151725
edited Jan 1 at 10:59
Bob
asked Dec 27 '18 at 8:28
BobBob
1,6151725
asked Dec 27 '18 at 8:28
BobBob
1,6151725
asked Dec 27 '18 at 8:28
BobBob
1,6151725
1,6151725
add a comment |
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